lysozyme partitioning in novel aqueous-two phase systems ... · lysozyme partitioning in novel...
TRANSCRIPT
Lysozyme Partitioning in Novel Aqueous-two PhaseSystems based on Hyperbranched Polymers
Beatriz Noriega Fernandes
Thesis to obtain the Master of Science Degree in
Biological Engineering
Supervisors: Prof. Tim ZeinerProf. Ana Margarida Nunes da Mata Pires de Azevedo
Examination Committee
Chairperson: Prof. Gabriel Antonio Amaro MonteiroSupervisor: Prof. Maria Raquel Murias dos Santos Aires Barros
Member of the Committee: Prof. Ana Margarida Nunes da Mata Pires de Azevedo
September 2015
Anyone who has never made a mistake has never tried anything new.Albert Einstein
Abstract
Aqueous two-phase systems (ATPS) constitute a promising process technology for the separation
and purification of biomolecules. Despite scalability and mild extraction conditions, commercial appli-
cation is scarce due to multiplicity of process variables, which lead to significant experimental efforts
towards the design of ATPS processes.
The aim of this work is to study and model the partitioning of lysozyme in ATPSs with linear and
hyperbranched polymers (HBP) as phase forming components. Dextran T40 and polyethylene glycol
(PEG) polymers with a molar mass of 6000 and 8000 were chosen as linear polymers. The used HBP
were hyperbranched polyester with a PEG core and 2,2-bis(methylol)propionic acid branching units.
The Lattice Cluster Theory (LCT) in combination with the Wertheim Association Theory was used
to model the studied ATPSs and the lysozyme partitioning.
The goal was partially achieved as the partitioning in the HBPs based ATPS could not be mea-
sured. Although, the ternary systems polymer – polymer – water were successfully modelled as well
as the lysozyme partitioning coefficients in the linear polymers’ based systems.
Keywords
Aqueous Two-Phase Systems (ATPS), Lysozyme, Lattice Cluster Theory (LCT), Whertheim The-
ory, Partition.
iii
Resumo
Os sistemas de duas fases aquosas (ATPS – aqueous two phase systems) constituem uma tec-
nologia promissora para a separacao e purificacao de biomoleculas. Apesar da possibilidade de au-
mento de escala e das condicoes de extraccao nao serem extremas para as proteınas, a aplicacao
comercial e ainda escassa devido a multiplicidade de variaveis de processo, o que tem levado ao au-
mento dos esforcos experimentais direccionados para o desenvolvimento de processos de ATPS. O
objectivo deste trabalho e estudar e modelar a particao da proteına lisozima em ATPS com polımeros
lineares e hiper-ramificados como componentes de formacao de fase. Os polımeros lineares escol-
hidos foram Dextrano T40 e dois tipos de polietileno glicol (PEG) com massas molares de 6000 e
8000. Os polımeros hiper-ramificados usados foram de poliester hiper-ramificado com um nucleo de
PEG e unidades ramificadas de acido 2,2-bis(metilol)propionico.
A Lattice Cluster Theory (LCT) em conjuncao com a Teoria de Associacao de Wertheim foi usada
para modelar os ATPS estudados e a particao da lisozima.
O objectivo foi so parcialmente atingido, uma vez que nao foi possıvel medir a particao nos ATPS
baseados em polımeros hiper-ramificados. Contudo, os sistemas ternarios polımero – polımero –
agua foram modelados com sucesso, assim como os coeficientes de particao da lisozima nos sis-
temas baseados em polımeros lineares.
Palavras Chave
Sistema de duas fases aquosas, Lisozima, Lattice Cluster Theory, Teoria de Wertheim, Particao.
v
Contents
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 State of the Art 5
2.1 Aqueous Two Phase System (ATPS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Hyperbranched polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Protein partitioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4 Fundamental Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.5 Modelling of ATPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.5.1 Empirical Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.5.2 Osmotic viral equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.5.3 Equations of state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.5.4 Activity coefficient models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.5.4.A Flory-Huggins Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3 Experimental Setup 19
3.1 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.1.1 Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.1.2 Lysozyme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.1.3 Solvents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Determination of the binodal curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3 Introduction to High Performance Liquid Chromatography . . . . . . . . . . . . . . . . . 22
3.4 HPLC method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.5 Calibration of the polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.6 Determination of Tie Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.7 Partitioning of the Lysozyme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4 Model 37
4.1 Lattice Cluster Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.2 Wertheim theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
vii
5 Results and Discussion 42
5.1 Liquid-liquid equilibria for the lysozyme-water system . . . . . . . . . . . . . . . . . . . . 43
5.2 Liquid-liquid equilibria for ternary systems . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.2.1 Calibration for tie-line determination . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.2.2 Tie-line modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.3 Partitioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.3.1 Calibration for lysozyme quantification . . . . . . . . . . . . . . . . . . . . . . . . 53
5.3.2 Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
6 Conclusions and Future Work 63
Bibliography 67
Appendix A Lattice Cluster Theory (LCT) correction terms A-1
Appendix B Experimental data B-1
B.0.3 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-1
B.0.4 Systems without lysozyme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-2
B.0.5 Systems with Lysozyme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-5
B.0.6 Partitioning data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-6
viii
List of Figures
2.1 Schematic ternary diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Number of publications containing ”Aqueous Two Phase Systems” and ”Protein” from
1970 to 2014. Date: 26th March 2015, sourse: www.scopus.com . . . . . . . . . . . . . 8
2.3 Schematic description of dendritic polymers. . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Equilibrium between two phases for a pure component. . . . . . . . . . . . . . . . . . . 12
2.5 Illustration of the distribution of chains of polymers and a solvent in a lattice. . . . . . . . 15
3.1 Idealized structure of PFLDHB-G2-PEG6k-OH (G2). . . . . . . . . . . . . . . . . . . . . 20
3.2 Idealized structure of PFLDHB-G3-PEG6k-OH (G3). . . . . . . . . . . . . . . . . . . . . 21
3.3 Flow Sheet of High-performance Liquid Chromatography (HPLC). . . . . . . . . . . . . 22
3.4 Example of a typical chromatography diagram. . . . . . . . . . . . . . . . . . . . . . . . 23
3.5 Chromatogram obtained from a PEG 6000 – Dextran aqueous solution at 80°C with a
flow rate of 0,2 ml/min. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.7 Dextran influence in the lysozyme Ultraviolet (UV) spectrum. . . . . . . . . . . . . . . . 32
3.8 Absorbance of lysozyme at 220 and 280 nm. . . . . . . . . . . . . . . . . . . . . . . . . 33
3.9 Influence of dextran concentration in the measurement of lysozyme concentration. . . . 33
4.1 United atom group models for monomers of poly(ethylene) (PE), poly(propylene) (PP),
poly(thylene propylene) (PEP) and poly(isobutylene). Circles represent CHn groups,
solid lines designate C - C bonds inside the monomer, and dotted lines indicate C - C
bonds linking the monomer to its neighbours along the chain. . . . . . . . . . . . . . . . 38
5.1 Lysozyme structure illustration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.2 Liquid-liquid equilibrium of aqueous lysozyme solution. . . . . . . . . . . . . . . . . . . . 44
5.3 Calibration Diagram for the linear polymers. . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.4 Calibration Diagram for the Hyperbranched Polymers (HBP). . . . . . . . . . . . . . . . 45
5.5 Experimental and LCT+Wertheim modelled points of ATPS consisting of linear polymers. 47
5.6 Experimental and LCT+Wertheim modelled points of ATPS consisting of HBP polymers. 50
5.7 Comparison can be made between the linear and the hyperbranched systems. . . . . . 51
5.8 Calibration diagram for the lysozyme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.9 Calibration diagram for G2 spectrometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.10 Experimental tie-lines of ATPS consisting of PEG 6000 – dextran 40 – water – lysozyme. 56
ix
5.11 Experimental tie-lines of ATPS consisting of PEG 8000 – dextran 40 – water – lysozyme. 56
5.12 Experimental tie-lines of ATPS consisting of HBP – dextran 40 – water – lysozyme. . . . 57
5.13 Lysozyme partition coefficient in PEG 6000 – Dextran and PEG 8000 – Dextran Aque-
ous Two Phase Systems (ATPS). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.14 Lysozyme partition coefficient in PEG 6000 – Dextran – water and PEG 6000 – Dextran
– water, experimental and calculated results. . . . . . . . . . . . . . . . . . . . . . . . . 59
5.15 Lysozyme partition coefficient in PEG 6000 – Dextran – water and PEG 6000 – Dextran
– water, experimental and calculated results. . . . . . . . . . . . . . . . . . . . . . . . . 60
x
List of Tables
3.1 Characteristics of the used polymers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 My caption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3 Retention time differences between PEG 6000 and dextran for different flow rates at 80
°C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.4 Summary of the methods and measurement times. . . . . . . . . . . . . . . . . . . . . . 27
3.5 Prepared bulk solutions for calibration of the polymers. . . . . . . . . . . . . . . . . . . . 27
3.7 Prepared PEG 8000 Samples for Calibration. . . . . . . . . . . . . . . . . . . . . . . . . 28
3.6 Prepared PEG 6000 Samples for Calibration. . . . . . . . . . . . . . . . . . . . . . . . . 28
3.8 Prepared Dextran Samples for Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.9 Prepared G2 Samples for Calibration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.10 Prepared G3 Samples for Calibration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.11 Composition and measured values of the components of the systems used to obtain
the tie lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.12 Information about apparatus and Software in spectrophotometry. . . . . . . . . . . . . . 31
3.13 Composition of the solutions analysed in Figure 3.7. . . . . . . . . . . . . . . . . . . . . 32
3.14 Prepared G2 samples for calibration in spectrophotometer. . . . . . . . . . . . . . . . . 34
3.15 Prepared solutions to test the lysozyme measurement method in samples containing
G2. BS stands for bulk solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.16 Composition of prepared systems with lysozyme and linear polymers. . . . . . . . . . . 35
3.17 Composition of prepared systems with lysozyme and HBP. . . . . . . . . . . . . . . . . 35
3.18 Prepared lysozyme samples for calibration in spectrophotometer. . . . . . . . . . . . . . 36
5.1 Binary interaction, association and LCT parameters for the aqueous lysozyme solution. 44
5.2 Association parameters of water and linear polymers. . . . . . . . . . . . . . . . . . . . 46
5.3 Interaction parameters of LCT+Wertheim for systems with linear polymers. . . . . . . . 47
5.4 Summary of LCT and Wertheim Association parameter set for linear polymers. . . . . . 48
5.5 Summary of LCT and Wertheim Association parameter set for the HBP. . . . . . . . . . 52
5.6 Prepared and calculated concentrations, measured and calculated absorbances, and
error fo the solutions used on the explained test. . . . . . . . . . . . . . . . . . . . . . . 54
5.7 Average mixing point of the partitioning experiments. . . . . . . . . . . . . . . . . . . . . 55
xi
5.8 Absorbance and lysozyme concentration on the top and bottom phases of ATPS con-
taining G2. The A correspond to systems with 0,3 wt % of lysozyme and the B to
systems with 1 wt %. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
B.1 Calibration data for the linear polymers. . . . . . . . . . . . . . . . . . . . . . . . . . . . B-1
B.2 Calibration data for the HBP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-2
B.3 Calibration data for lysozyme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-2
B.4 Calibration diagram for G2 spectrometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . B-2
B.5 Experimental data of ATPS system PEG6000 – dextran – water, for tie-line 1. . . . . . . B-2
B.6 Experimental data of ATPS system PEG6000 – dextran – water, for tie-line 2. . . . . . . B-3
B.7 Experimental data of ATPS system PEG6000 – dextran – water, for tie-line 3. . . . . . . B-3
B.8 Experimental data of ATPS system PEG8000 – dextran – water, for tie-line 1. . . . . . . B-3
B.9 Experimental data of ATPS system PEG8000 – dextran – water, for tie-line 2. . . . . . . B-3
B.10 Experimental data of ATPS system PEG8000 – dextran – water, for tie-line 3. . . . . . . B-3
B.11 Experimental data of ATPS system G2 – dextran – water, for tie-line 1. . . . . . . . . . . B-4
B.12 Experimental data of ATPS system G2 – dextran – water, for tie-line 2. . . . . . . . . . . B-4
B.13 Experimental data of ATPS system G3 – dextran – water, for tie-line 1. . . . . . . . . . . B-4
B.14 Experimental data of ATPS system G3 – dextran – water, for tie-line 2. . . . . . . . . . . B-4
B.15 Experimental data of ATPS system G3 – dextran – water, for tie-line 3. . . . . . . . . . . B-4
B.16 Experimental data of ATPS system PEG6000 – dextran – water with 0,3 wt% lysozyme,
for tie-line 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-5
B.17 Experimental data of ATPS system PEG6000 – dextran – water with 0,3 wt% lysozyme,
for tie-line 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-5
B.18 Experimental data of ATPS system PEG6000 – dextran – water with 1 wt% lysozyme,
for tie-line 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-5
B.19 Experimental data of ATPS system PEG6000 – dextran – water with 1 wt% lysozyme,
for tie-line 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-5
B.20 Experimental data of ATPS system PEG8000 – dextran – water with 0,3 wt% lysozyme,
for tie-line 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-5
B.21 Experimental data of ATPS system PEG8000 – dextran – water with 0,3 wt% lysozyme,
for tie-line 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-6
B.22 Experimental data of ATPS system PEG8000 – dextran – water with 1 wt% lysozyme,
for tie-line 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-6
B.23 Experimental data of ATPS system PEG8000 – dextran – water with 1 wt% lysozyme,
for tie-line 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-6
B.24 Experimental lysozyme concentration used to calculate the partition coefficients for
systems with PEG 6000 - first measurement. . . . . . . . . . . . . . . . . . . . . . . . . B-6
B.25 Experimental lysozyme concentration used to calculate the partition coefficients for
systems with PEG 6000 - second measurement. . . . . . . . . . . . . . . . . . . . . . . B-6
xii
B.26 Experimental lysozyme concentration used to calculate the partition coefficients for
systems with PEG 8000 - first measurement. . . . . . . . . . . . . . . . . . . . . . . . . B-7
B.27 Experimental lysozyme concentration used to calculate the partition coefficients for
systems with PEG 8000 - second measurement. . . . . . . . . . . . . . . . . . . . . . . B-7
B.28 Experimental partition coefficients and deviation for the ATPS formed by linear polymers.B-7
xiii
Abbreviations
ATPS Aqueous Two Phase Systems
bis-MPA 2,2-bis(methylol)propionic acid
EOS Equation of State
FH Flory-Huggins
G2 PFLDHB-G2-PEG6k-OH
G3 PFLDHB-G3-PEG6k-OH
HBP Hyperbranched Polymers
HPLC High-performance Liquid Chromatography
LCT Lattice Cluster Theory
LLE Liquid-Liquid Equilibrium
NRTL Non Random Two Liquids
PC-SAFT Perturbed-chain Statistical Associating Fluid Theory
PEG Polyethylene Glycol
PFC Phase Forming Components
RID Refractive Index Detector
SAFT Statistical Association Fluid Theory
SEC Size Exclusion Chromatography
TL Tie Line
TLL Tie Line Length
UNIQUAC Universal Quasichemical
UV Ultraviolet
VERS Viral Equation Relative Surface
VLE Vapour-Liquid Equilibrium
xv
Nomenclature
Latin Letters
Acalc Calculated absorbanceAest Estimated absorbanceAmeas Calculated absorbanceb3,i Number of branching points of three of component
ib4,i Number of branching points of four of component i
∆E1 Energy contribution of the first order∆E2 Energy contribution of the second order
∆mixG Gibbs free energy of mixtureG Total Gibbs free energyg Gibbs energy per segmentGE Excess Gibbs free energykB Boltzmann constantKP Partition coefficientKassoij Association volume defined by Eq. (4.8)
MM i Molar mass of the component iNi Number of molecules of the component ini Amount of substance of component iNL Number of lattice sitesP PressureR Ideal gas constantS Combinatorial entropy of a mixtureMi Number of segments of the component iT TemperatureA AbsorbanceT Transmittance
XSi X
εi X
ε2
i coefficients defined by eqs. (A.1) to (A.5),eqs. (A.6) to (A.11), eqs. (A.12) to (A.18)
xi Mole fraction of the component iz Coordination number
Greek Letters
∆ij association strength between molecule i andmolecule j defined by Eq. (4.7)
∆εij Interaction energy defined by (2.11)εassoij Association energy between segments of molecule
i and molecule j.Φi Segment fraction of component iµ Chemical PotentialΩ Number of possible configurations on a lattice
xvii
1Introduction
Contents1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1
1.1 Motivation
Over the last 20 years a great productivity increase took place in the biomanufacturing industry.
Titers of recombinant proteins in fermentation broths were boosted from a scale of milligrams to
grams per litre [1]. This development was especially motivated by the increasing market value of
biopharmaceutical products, particularly antibodies and proteins [2]. Simultaneously, downstream
processing did not keep up with the productivity improvement, despite still being the main responsible
for the production cost of biopharmaceuticals (between 45% and 92% [3]), thus creating a bottleneck
in the biotechnological industry [4].
Notably, therapeutic proteins require high purification values (<0,1% impurities [5]). Proteins de-
rive their functions from a complex three-dimensional structure that can easily be lost when subjected
to pH and temperature values far from moderate. Moreover, structural damage can be irreversible,
leading to biomolecules without functionality, and therefore, value. Chromatography is a simple pro-
cess with a high resolving power that embodies the right conditions for the purification of biomolecules.
For that reason it becomes indispensable as a unit operation. However, it has low capacity and is dif-
ficult to scale up, hence being one of the reasons for the high costs of downstream processing [6].
In an attempt to either replace chromatography or reduce the load of impurities in the feedstream so
that one or more chromatography stages can be eliminated, alternative separation processes have
been proposed. Aqueous Two Phase Systems (ATPS) show potential to overcome the limitations of
chromatography.
ATPS are formed by mixing two hydrophilic, incompatible components, such as a salt and a poly-
mer, a salt and an alcohol, two hydrophilic polymers or two surfactants in water above a critical con-
centration [7]. These systems can be used effectively for the separation and purification of proteins
given that both phases consist mainly of water, and have a low interfacial tension resulting in a low
mechanical stress on the proteins during the extraction process. The practical application of ATPS
has been demonstrated on a large number of cases with excellent levels of purity and yield [8].
Separation processes based on ATPS overcome chromatography limitations: it is possible to run
them continuously [9] and they can easily be scaled-up [10]. Consequently, ATPS have been the
object of extensive studies regarding the purification of therapeutic proteins and others of industrial
interest [11], although industrial applications are still rare.
Nonetheless, these systems present some disadvantages. The polymer-salt systems are not opti-
mal due to the low solubility of amphiphilic molecules in the salt phase - high tendency of aggregation
[12] - and the shielding of proteins by the salt, which prevents an easy adsorption in further necessary
separation steps [13]. Furthermore, salt-polymer systems are generated with a high amount of salt,
and thus require a large consumption of phase-forming chemicals [14], which gives rise to environ-
mental and economical problems. On the other hand, polymer-polymer systems do not present the
listed problems, but do however have high viscosities, which is a problem in the case of industrial use
[14].
As a possible solution to the mentioned problems with ATPS, Kumar et al. [15] suggested the use
2
of smart polymers. These are characterized by their ability of undergo large reversible chemical and
physical changes as a response to small environmental variations. Such molecules can be sensitive
to factors such as temperature, pH, electric or magnetic field and light intensity [16]. One specific
type of such materials are the Hyperbranched Polymers (HBP) – compounds that can carry special
functional groups and show no denaturation effect on the biomolecules. HBP can be used as phase
forming components in ATPS originating systems with lower viscosity and melting point, and higher
specificity [13]. Application of HBP in ATPS has shown promising potential [15], which encourages
the study of these systems.
The design of an extraction using ATPS is a time consuming process that requires enormous
experimental efforts in order to characterize the behaviour of each component in the mixture. In
addition, when a system has low productivity, it is necessary to restart the process. Thus, for the
optimization of extraction processes with ATPS, a modelling of these systems is required.
Therefore, this work will focus on answering three main questions:
1. Can lysozyme be purified with aqueous polymer-polymer systems composed by linear PEG
6000, 8000 and Dextran?
2. Can the lysozyme partitioning be modelled with an Lattice Cluster Theory (LCT) model in asso-
ciation with the Wertheim theory?
3. Can systems composed by HBP also be modelled accurately?
The modelling will be based on a version of the LCT applicable for multicomponent polymer solu-
tions combined with the Wertheim theory so as to account for the associative interactions [13].
1.2 Thesis Outline
Firstly, the state of the art of the studied field is presented. An introduction to aqueous two phase
systems is made, followed by a summary of hyperbranched polymers’ history, their applications and
importance. Ensuing, the protein partitioning behaviour in aqueous two phase systems is briefly
explained. The fundamental thermodynamics of these systems are also presented. Finally the state
of the art on aqueous two phase systems’ modelling is summarized.
The experimental setup of this work is then described. The used materials are discriminated and
characterized. The high-performance liquid chromatography and spectrophotometry methods are
explained in detail, as are the experimental procedures followed in the analysis. In this section the
choice of method is also explained.
The modelling chapter explains, in a summarized way, the applied models, Lattice Cluster Theory
(LCT) and Wertheim Theory, and presents the correspondent equations.
Afterwards, the final results are presented. Aqueous two phase systems containing two linear
PEG polymers, dextran and two hyperbranched polymers are studied experimentally and adjusted
to a LCT+Wertheim theory model. Furthermore, the lysozyme partitioning on the linear polymers
3
based systems is measured and adjusted to a quaternary LCT+Wertheim theory model. The obtained
results are analysed and discussed.
Finally, the last chapter contains the conclusions of the work and suggests future developments
on the topic.
The work behind this thesis was performed in the Fluid Separation Laboratory of the Department
of Biochemical and Chemical Engineering in the Technical University of Dortmund, Germany.
4
2State of the Art
Contents2.1 Aqueous Two Phase System (ATPS) . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Hyperbranched polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Protein partitioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4 Fundamental Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.5 Modelling of ATPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
5
In an attempt to describe the phase equilibria in ATPS, different models were developed, which
can be segregated based on divergent schools of thought. The first modelling approaches were
based on the Othmer-Tobias [17] and Bancroft [18] equations, which required an input of experimental
data and several adjusted parameters to model the tie-lines. Other models are based on osmotic
viral expansions, reminiscent of the work of Edmond and Ogston [19][20]. These do not require
experimental data input, but involve up to three adjustable parameters to describe the interaction
between the different components, and an extra one to account for the temperature influence.
Besides the correlation equations, thermodynamic models using Equation of State (EOS) and
the Gibbs excess energy (GE) equations where also introduced to perform calculations in order to
characterize the ATPS. One possible way is the use of EOS, such as Statistical Association Fluid
Theory (SAFT)-family equations, which originated from a perturbation theory inspired by Wertheim
papers [21]. The EOS models are useful when modelling different phase diagrams with polymers
present in the mixture, and normally pure component data are used to fit the model parameter of an
EOS as density or vapour pressure, but depend upon high calculation efforts and the parameters of
pure components must be fitted to phase equilibria data.
The GE models have also been applied to calculate the phase behaviour of ATPS. These models
can take into account short range interactions, such as the Non Random Two Liquids (NRTL), ex-
tended NRTL, Wilson or Universal Quasichemical (UNIQUAC) models, or long range interactions, in
the case of Debye-Hueckle and Pitzer-Debye-Hueckle [22]. The drawback of the GE models is the
lack of associative interactions which are of great importance for the thermochemical modelling of
liquid systems.
Lastly, there are also models based on extensions of lattice theories (also GE models) such as
the Flory-Huggins (FH) theory [23] [24], which is the classical way of describing phase behaviour in
polymer solutions. However this theory is a simple mean-field approximation that ignores the details
of the polymer structure, and therefore, is not suitable for non-linear polymers. The LCT is an exten-
sion of the FH theory, developed by Freed and co-workers [25–29], that takes into account polymer
structure.
Kulaguin Chicaroux and Zeiner [13] published a GE model based on the LCT to calculate the
ATPS taking into account on the one hand the molecular structure, derived from the LCT, and on the
other hand the associative interactions by integrating the Wertheim association theory [30, 31].
2.1 Aqueous Two Phase System (ATPS)
As mentioned before, ATPS are formed by mixing two hydrophilic, incompatible components, such
as a salt and a polymer, a salt and an alcohol, two hydrophilic polymers or two surfactants in water
above a critical concentration [7]. In most cases, both aqueous phases contain 50 to 90 wt% of
water, and the type and concentration of the components in the mixture is a decisive factor for phase
formation. Physical properties such as density, interfacial tension or viscosity of phases or between
them have a strong dependence on the Phase Forming Components (PFC).
6
Generally the Liquid-Liquid Equilibrium (LLE) of an ATPS at constant temperature and pressure
is described by a ternary diagram, exemplified in Figure 2.1. With one point within the triangular
diagram, the composition in mass fraction or mole fraction can be read out. In the diagram, the curve
represents the binodal curve which separates the immiscible region (grey) from the homogeneous
one-phase region (white). Inside the grey region all the mixing compositions correspond to a tie-line
(broken line) and originate a system with two phases in equilibrium, each with the composition given
by the point where the tie-line intercepts the binodal curve. At the critical point (•), the tie line length
is zero. The mass fraction or mole fraction of every component in the two phases is equal.
Figure 2.1: Schematic of a ternary diagram. The solid line represents the binodal curve of the system andbroken lines represent the tie-line. The white region represents the compositions that form a miscible one-phasemixture and the grey region represents the compositions that form an heterogeneous two-phase mixture. The
full circle (•) corresponds to the critical point.
A possible approach to characterize and study ATPS is the Tie Line Length (TLL), which is used
to compare the physicochemical differences between the two phases [7]. In an aqueous two-phase
diagram, a Tie Line (TL) is a straight line between two points on the binodal curve, connecting the
two corresponding phases (top and bottom phase concentration) for a specific mixing point. It can be
calculated using the Pythagoras theorem, (2.1).
TLL =√
(C1,TP − C1,BP )2 + (C2,TP − C2,BP )2 + ...+ (Cn,TP − Cn,BP )2 (2.1)
Where 1,2...n are the PFC of the mixture, and Ci,TP and Ci,BP correspond to the concentration of
the component in the top and the more dense bottom phases, respectively. The binodal curve of an
ATPS changes with different factors such as the type of the PFC, the type of additional solutes used,
or the temperature. As a consequence, the TL is also affected by these parameters, not only in its
length but also in its slope. Additionally, the TLL can also be used to calculate the phase ratio of the
systems with the lever rule, knowing the mixing point [32].
The first polymer-polymer ATPS publication was in 1896 by Beijerinck [33], which described the
7
formation of an emulsion after mixing a starch solution with a gelatin solution. The first polymer-
salt ATPS was discovered by accident by Per-Ake Albertsson [34]. After the discovery, in the 50s
Albertsson studied the partitioning of proteins and cell particles in ATPS of polymer-polymer and
polymer-salt [35][36], and in the following years he also studied low molecular weight substances,
polyelectrolytes, nucleic acids, and membrane vesicles, which are non-soluble compounds [37].
Consecutive to his research, many groups continue investigating the ATPS application in biomolecule
purification with focus in different topics such as factors influencing the scale-up, the partitioning, or
process variants [38] [7] [39].
The growing number of publications regarding ATPS and proteins since 1970 can be observed in
Figure 2.2.
1 9 7 0 1 9 7 5 1 9 8 0 1 9 8 5 1 9 9 0 1 9 9 5 2 0 0 0 2 0 0 5 2 0 1 0 2 0 1 50
2 5
5 0
7 5
1 0 0
1 2 5
Numb
er of
public
ation
s
Y e a r
Figure 2.2: Number of publications containing ”Aqueous Two Phase Systems” and ”Protein” from 1970 to 2014.Date: 26th March 2015, sourse: www.scopus.com
As can be seen in Figure 2.2, the number of publications shows a steady increase, and since the
number of publications correlates to the research effort put into this topic, it is safe to assume that
the interest is still growing. This strong increase can be partly explained by the more recent study of
the application of ATPS for the purification of biopharmaceuticals and biocatalysts by several groups,
such as the ones of Aires-Barros [40], Rita-Palomares [41] and Asenjo [42].
2.2 Hyperbranched polymers
In the end of the 19th century, Berzelius reported the synthesis of a resin from tartaric acid and
glycerol – the first reported synthesized hyperbranched macromolecule [43]. In 1909, Baekeland
introduced the first commercial synthetic plastics, phenolic polymers, commercialized through his
Bakelite Company [44]. In the 1940s, Flory et al. [45][46][47] developed the ”degree of branching”
8
and ”highly branched species” concepts, but the term ”Hyperbranched Polymer” was first used in 1988
by Kim and Webster [48].
HBP are highly branched macromolecules with three-dimensional dendritic architecture. Due to
their unique physical and chemical properties and potential applications in various fields from drug-
delivery to coatings, interest in HBP is growing rapidly [49].
HBP are one type of dendritic polymers, which are the fourth major class of polymers’ architecture,
and consist of six subclasses [50], illustrated in Figure 2.3:
1. dendrons and dendrimers;
2. linear-dendritic hybrids;
3. linear-dendritic hybrids
4. dendrigrafts or dendronized polymers;
5. hyperbranched polymers;
6. multi-arm star polymers;
7. hypergrafts or hypergrafted polymers.
Figure 2.3: Schematic description of dendritic polymers [49].
What distinguishes these groups is the degree of structural control during the synthesis. HBP are
the most irregular dendritic polymers.
The main differences between linear and HBP are the glass temperature (temperature at which
occurs a reversible transition in amorphous materials from a hard and relatively brittle state into a
molten or rubber-like state), the viscosity and the solubility.
9
The glass temperature is a function of the backbone, the number of end-groups, and the number
of branching points. The glass temperature lowers with the increase of end-groups, while it increases
with the number of branching points. Markedly, the influence of end-groups for a linear polymer
of infinite molar mass disappears while for dendrimers it converges to a certain value, if the glass
temperature is affected by the polarity of end-groups [51].
One of the most interesting physical properties of HBP is their considerably different viscosity
characteristics in comparison with their linear analogues. Hyperbranched macromolecules in solution
reach a maximum of intrinsic viscosity as a function of molecular weight as their shape changes,
however the conditions for the existence of this maximum are still not clear [52]. Moreover, the melt
viscosity of linear polymers has been shown to increase linearly with logarithm of molar mass up to
a critical molar mass where the viscosity drastically increases [53]. This behaviour is related to the
enlargement of the polymer chain with the molar mass, which has been proven not to occur in the
case of the HBP [54].
The solubility of dendritic polymers with a higher generation (number of repeated branching cy-
cles that are performed during its synthesis) depend largely on the characteristic of the end groups,
therefore, modifying these groups permits to control influence of their solubility [55].
Due to their unique properties and easy synthesis, HBP have a wide range of potential appli-
cations, from being used as nanomaterials for host-guest encapsulation [56], to the fabrication of
organic-inorganic hybrids [57], as nanoreactors, or as biocarriers and biodegradable materials [49].
Some other application examples are their use as rheology modifiers or blend components [58], as
the base for various coating resins, or as novel polymeric electrolytes or ion-conducting elastomers
[49].
Different models have been proposed to predict thermodynamic properties and phase equilibria
of mixtures containing hyperbranched or dendritic polymers. The branched Perturbed Chain Polar-
Statistical Association Fluid Theory model was used to calculate phase equilbria of HBP by Kozlowska
et al. [59]. Lattice Cluster Theory (LCT) was used to calculate the liquid-liquid phase equilibria of
HBP solutions and to determine the activity coefficients of dendrimer solutions [60][61]. UNIFAC-FV
(Universal Quasichemical Functional Group Activity Coefficients- Free Volume) approach was also
tested to calculate vapour-liquid phase equilibria of HBP solution [62].
2.3 Protein partitioning
Proteins can either be separated from cell debris or purified from other proteins using two-phase
partitioning. Most soluble and particulate matter will partition to the lower, more polar phase, whilst
proteins will partition to the top, less polar and more hydrophobic phase, usually Polyethylene Glycol
(PEG) [11]. The partition coefficient can be manipulated in order to separate proteins from one
another. It is influenced by the average molecular weight of the polymers, the type of ions in the
system, the ionic strength of the salt phase or addition of an additional salt such as NaCl. Affinity
partitioning, a method for the purification of proteins containing specific ligand binding receptor sites,
10
can also be used to increase the degree of purification [63].
The partition coefficient, KP , can be calculated with Equation (2.2).
KP =Ci,BPCi,TP
(2.2)
Where Ci,TP and Ci,BP correspond to the equilibrium concentration of the protein in the top and
bottom phases, respectively.
The mechanism governing partition can be described qualitatively as follows: a protein interacts
with the surrounding molecules within a phase via various types of interactions, such as hydrogen,
ionic, and hydrophobic interactions, together with other weak forces. The net effect of these interac-
tions is likely to be different in the two phases and therefore the protein will preferentially partition to
one phase.
The properties of proteins, such as hydrophobicity, electrochemical interactions, molecular size,
shape and surface area, biospecific affinity and conformation, can be exploited individually or in con-
junction to achieve an effective separation [64][11]. Thus, the overall partition coefficient can be
expressed in terms of all these individual factors:
KP = K0 ·Khfob ·Kel ·Kbiosp ·Ksize ·Kconf (2.3)
where hfob, el, biosp, size and conf stand for hydrophobic, electrochemical, biospecific, size and
conformational contributions to the partition coefficient, and K0 includes other factors.
In a polymer-polymer ATPS there are factors that also influence the partitioning behaviour, namely
the pH of the mixture, the molecular weight of the polymers, their concentration and the addition of a
salt such as NaCl [65] [11].
Specifically, the larger the molecular weight of the PEG, the lower the value of the partition co-
efficient [11]. Boncina et al. [66] studied lysozyme solubility in PEG solutions and observed that it
decreases linearly with the increase of PEG concentration. Furthermore, the solubility is lower for a
polyethylene glycol with a higher degree of polymerization. The solubility decrease is correlated with
the hydrophobic character of PEG molecules, which increases as the polymer’s molecular mass does
[67].
2.4 Fundamental Thermodynamics
Considering the systems of Figure 2.4 with two phases, α and β, and n pure components, the
equilibrium is reached for specific thermal, mechanical and chemical conditions described by the
Equations (2.4) to (2.6) [68].
11
µiα
µiβ
α
β
P T
Figure 2.4: Equilibrium between two phases for a pure component.
Tα = T β (2.4)
Pα = P β (2.5)
µαi = µβi ∀ i = 1, ..., n (2.6)
The equilibrium of the two phases is only possible if a list of requirements is fulfilled. For one, both
phases must share the same temperature – thermal equilibrium (2.4). Secondly, the pressure of the
phases must be the same – mechanical equilibrium (2.5). Lastly there must be equilibrium of matter,
i.e., the transition of mass between phases has to be the same and, as a result, the composition of
each component will remain constant in each phase (2.6).
Therefore, to determine the thermodynamic equilibrium of the system the chemical potential of
each component must be known, and for real solutions it is defined as shown in Equation (2.7).
µi =
(∂G
∂ni
)P,T,nj 6=i
(2.7)
Equation (2.7) shows that once the Gibbs energy is known, the chemical potential can be calcu-
lated. Several different models, either empirical or based on theories of the liquid state, have been
developed since the 19th century aiming to calculate the excess Gibbs free energy. These models will
be introduced on Chapter 2.5.
The choice of the model to use in each situation is a complex problem that requires a careful analy-
sis of the chemical and structural nature of the mixture components, the application, the experimental
data available, among others.
2.5 Modelling of ATPS
2.5.1 Empirical Equations
The empirical equations for calculating equilibrium of three components on two phases are divided
into two main groups, the Othmer-Tobias equation [69] and the Bancroft equation [70]. This method
has the advantage of combining the simple applicability of an empirical approach to a high precision.
12
Nevertheless, the use of these equations carries some drawbacks. For one, they can only be
applied to systems with three components, and since a single equation is being used to calculate the
composition of three components in two phases, plus two empirical parameters, the binodal curve
of the system must be known. The binodal curve can also be calculated via empirical equations,
however three or four additional empirical parameters are needed [71][72]. As the parameters do not
require any thermodynamic background, they must be adjusted for each new condition, which is why
empirical models cannot be used for prediction.
2.5.2 Osmotic viral equation
Although different versions of the osmotic viral equation can be applied to model ATPS, the first
adaptation by Edmond and Ogston to PEG-dextran systems in 1968 is the most popular due to its
simplicity and accuracy [19]. This equation is based on the osmotic viral expansions from McMillan
and Mayer, developed in 1945 [73].
The key ideas of the osmotic viral equation are, firstly, that the chemical potential of a solute can
be described by a Taylor series expansion of the solute concentration; secondly, that the chemical
potential of a solvent can be calculated with all the solutes’ potentials and the Gibbs-Duhem equation.
For an ATPS equilibrium with one solvent, water, and two solutes, three parameters are necessary:
two binary osmotic viral coefficients of the solutes, obtained by fitting to solvent activities, plus one
osmotic cross viral coefficient, obtained by fitting to LLE data. It is usual to fit the three mentioned
parameters only to LLE data, ignoring the equilibrium, and for that reason the viral coefficients that
are published only apply to LLE.
In the same manner as the empirical equations introduced in Section 2.5.1, the osmotic viral
equation cannot be used as a predicting tool, since the viral osmotic coefficients must be defined for
each system condition.
2.5.3 Equations of state
By using equations of state to model ATPS, the phase composition and volumetric properties can
be obtained with the model, which allows for an easier planning of the extraction process, as the
density of both phases influences the phase separation. Also, this method is able to predict without
interaction parameters or experimental data.
The first application of equations of state for ATPS modelling was made by Naeem and Sad-
owski (47), when they employed polyelectrolyte Perturbed-chain Statistical Associating Fluid The-
ory (PC-SAFT) for a system of polymer-polyelectrolyte and salt. Secondly, Valvary et al. [74] applied
the hard sphere chain for systems containing PEG and salts. This application in particular requires
an enormous amount of binary parameters which are functions of the molecular weight of the poly-
mer, of temperature, and even of the specific experimental-data reference. Moreover every salt was
described with its own parameter set, which again increased the amount of required parameters. As
a consequence, this specific method cannot yet be used as predictive model [75].
13
2.5.4 Activity coefficient models
In 1895, Max Margules modelled the Gibbs free energy of a solution with an empirical approach
[76], and since then, many activity coefficient models have been developed for modelling phase prop-
erties. In 1985, the first activity coefficient model for an ATPS was Flory-Huggins (FH) solution theory
[23] [24] by Walter et al. [39]. Later, in 1988 UNIQUAC was applied for ATPS with two polymers [77].
Viral Equation Relative Surface (VERS) was applied by Grossmann et al. [78] to model ATPS with
PEG and dextran in 1993. In 2002, Zafarani-Moattar and Sadeghi [79] obtained a novel expression
with three terms: short-range, combinatorial and long-range electrostatic interactions term, by com-
bining a model developed by Chen with Debye-Hueckel [80]. The local composition approach has
been widely used, with several adaptations for different short-range interactions (UNIQUAC , Wilson
[81]).
The main advantage of this approach is its high accuracy when modelling aqueous solutions,
electrolyte solutions, and ATPS of water with two solutes. As a result of considering short-range,
long-range and combinatorial terms in these models, it is possible to predict systems with changing
molecular weights [81].
As a limitation, it should be considered that different temperatures can only be taken into ac-
count when using activity coefficient models if different interaction parameters are added for each
temperature [81]. Since local composition concepts are used in these calculations, it is necessary to
experimentally determine the volumetric characteristics of the phase forming components. Further-
more, the complexity of the model increases with the number of parameters because of the need to
derive the equations adapted to the local composition model.
In the following Section 2.5.4.A the activity coefficient model, Flory-Huggins (FH) Theory, will be
presented.
2.5.4.A Flory-Huggins Theory
The Raoult’s law of ideal solutions states that the vapour pressure of a solvent above a solution is
equal to the vapour pressure of the pure solvent at the same temperature scaled by the mole fraction
of the solvent present [82]:
Psolution = XsolventPosolvent (2.8)
This law was proven unsuitable to describe polymer solutions when the experimental data of
Vapour-Liquid Equilibrium (VLE) did not fit the predictions [83]. For this reason, in an attempt to
better describe polymer solutions, in the early 40s, Flory and Huggins [83, 84] developed individually
a simple lattice theory, that would come to be known as the Flory-Huggins theory.
This lattice theory 2.5 was presented with the following assumptions:
1. a quasi-solid (properties between solid and liquid) lattice in the liquid,
2. polymer segments and solvent molecules’ interchangeability in the lattice cells,
14
3. independence on composition,
4. no polydispersity (all polymer molecules are assumed to be the same size) and
5. concentration of polymer in a cell neighbourhood considered to be equal to the all-over average
concentration.
Pure polymer Pure solvent Homogeneous solution
Figure 2.5: Illustration of the distribution of chains of polymers and a solvent in a lattice.
In order to calculate the Gibbs free mixing energy, it is necessary to estimate a residual part
(∆Gres) and a combinatorial part (∆Gcomb) in (2.9).
∆Gres + ∆Gcomb = (∆H − T∆Sres)− T∆Scomb (2.9)
where ∆Sres corresponds to the residual mixing entropy and ∆Scomb to the combinatorial mixing
entropy. In the FH theory, the residual portion is given purely by enthalpic contributions (∆H), and
the combinatorial portion is estimated by the positioning of the molecules on the lattice (∆Scomb).
In order to explain the basic concepts of this theory, the calculations for a binary mixture will be
described as an example.
Residual contribution
For estimating the mixing enthalpy,
∆Gres = ∆H =∆ε12z
2Φ1Φ2 (2.10)
the interaction energy ∆ε12 is defined as:
∆ε12 = ε11 + ε22 − 2ε12 (2.11)
where ε11 and ε22 describe the interaction between segments of the same type, polymer and
solvent accordingly, and ε12 describes the interaction energy between solvent and polymer.
In equation (2.10), Φi is the segment fraction of each component, defined as
Φ1 =N1M1
NLΦ2 =
N2
NL, NL = N1M1 +N2 (2.12)
where Mi is the chain length of the component i, Ni is the number of molecules of component i,
and NL is the total number of lattice sites. z is the coordination number of the lattice, i.e., the number
15
of lattice sites adjacent to each segment, which was derived by Flory [23] and Huggins [24] with the
assumption:
z →∞ and ∆ε→ 0 ⇒ ∆εz
2→ finite (2.13)
It is common to write this parameter as the FH parameter χ:
χ =∆εz
2(2.14)
Combinatorial contribution
In a lattice, the molecules can be distributed and redistributed, and it is possible to estimate the
combinatorial part of the Gibbs free mixing energy with the Boltzmann relation, Equation (2.16), and
the number of all possible configurations of the system, Ω, neglecting molecule interaction, with kB
being the Boltzmann constant.
The entropy of mixing for a binary mixture corresponds to:
∆S = Smix − S1 − S2 (2.15)
in which Smix, S1 and S2 are the entropy of the mixture, and components 1 and 2. Each of these
values can be determined through Equation (2.16).
S = kB lnΩ (2.16)
Since components 1 and 2 are the same size and each molecule occupies one single lattice, the
number of possible dispositions in each case is only one, therefore:
Ω = 1→ lnΩ = 0→ S1 = S2 = 0 (2.17)
This is not the case for the entropy of the mixture. In a binary mixture, if the solvent and solute are
the same size, Ω can be calculated as is shown in Equation (2.18).
Ω =N !
N1!N2!(2.18)
with Ni being the number of molecules of component i in the mixture, and N the total number of
molecules.
N = N1 +N2 (2.19)
Inserting this calculation in the Boltzmann relation,
S = kB [ln(N !)− ln(N1!)− ln(N2!)] (2.20)
and resorting to the Stirling approximation:
ln(x!) = xln(x)− x (2.21)
16
the combinatorial contribution can be calculated as:
∆Scomb = −kB[N1ln
(N1
N
)+N2ln
(N1
N
)](2.22)
and, finally, considering the mole fraction,
xi =NiN
(2.23)
it can be expressed by Equation (2.24).
∆Scomb = −R [x1ln (x1) + x2ln (x2)] (2.24)
Since in a polymer solution the sizes of solute and solvent have a large difference, Flory [23] and
Huggins [24] introduced the idea of dividing the polymers into pieces – chain segments. Considering
that the solute molecules and the created segments have the same size, it is then possible to use the
Boltzmann relation to calculate the combinatorial entropy.
The number of segments, M , depends on both the polymer and the solvent molecular sizes, and
can be calculated with Equation (2.25).
M =MM polymer
MM solute(2.25)
where MM polymer and MM solute are the molar masses of the polymer and solute respectively.
The last required step to be able to apply the Boltzmann relation to a polymer solution is consider-
ing the strong bonds between the individual segments of the polymers, i.e., the polymer disposition in
the lattice should be in a way that all the segments have at least two neighbour segments, excluding
the first and the last, which only need to have one. Therefore, for polymer solutions, considering once
again the definition (2.15), the entropy of the pure polymer is no longer zero (Ω > 1), and the entropy
of the mixture must be calculated with a different expression.
The derived expression for the combinatorial part of the entropy of mixing can be found in Equation
(2.26).
∆Scomb = −k[N1ln
(MN1
N2 +MN1
)+N2ln
(N2
N2 +MN1
)](2.26)
Applying the definition of segment fraction (2.12), the segment mixing entropy can be written as
∆Scomb = −R[
Φ1
Mln(Φ1) + Φ2ln(Φ2)
](2.27)
Finally, the FH mixing energy is:
∆G = −RT[
Φ1
Mln(Φ1) + Φ2ln(Φ2)
]+ χΦ1Φ2 (2.28)
FH and related theories are still widely used for the thermodynamic description of polymer solu-
tions because they present several advantages. While classic FH theory successfully explains the
17
behaviour of long-chain polymers in the liquid state and gives mechanistic insight into the phase
formation process [85], it is inadequate in several respects.
There is a concentration range in which the FH theory is expected to be applied, due to the mean
field approximation used as the basis of the calculations. On the one hand, there is a minimum
concentration from which the solution can be considered to have a uniform density of segments; on
the other hand, given the lack of consideration for the interactions between non-successive polymer
segments, the solution can also be sufficiently diluted so that a polymer segment is approximately
surrounded only by solvent, and is therefore separated from its polymer neighbours.
Futhermore, due to the employment of a simple mean-field approximation that effectively ignores
the details of the polymer chain connectivity, it cannot distinguish between linear, star, branch, and
other polymer architectures [29].
The FH expression for the χ parameter, Equation (2.14), provides little further insight into the
physical features that control polymer miscibility. Actually, χ is described as inversely dependent on
the temperature, as well as independent of the composition, the molecular weights, and the pressure
of the mixture. However, χ is often found to be empirically dependent on the last three parameters,
and has shown a dependence on temperature better described by the relationship χ = A+B/T [29],
consequence of the non-combinatorial contributions not considered (volume, equation of state, and
packing effects). The contradictions between theory and experimental data implies that, although FH
is applicable in specific situations, the χ parameter is purely empirical and is inadequate for more am-
bitious predictive purposes. In the literature, several extensions of the FH theory can be found that try
to describe the χ-parameter depending on concentration and temperature, such as the Koningsveld-
Kleintjens Model and the Prigogine-Flory-Patterson Theory [86], [87].
FH specifies that monomers of different polymers and solvent molecules all occupy a single lattice
site, ignoring the monomer structure, e.g. the molecular size and shape, and as a consequence, the
influence on the mixture thermodynamics [29]. Moreover it does not consider the excess volume.
In order to correct some of the mentioned deficiencies, Freed and co-workers developed the LCT,
based on FH theory, that will be introduced in the following Section.
18
3Experimental Setup
Contents3.1 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2 Determination of the binodal curves . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3 Introduction to High Performance Liquid Chromatography . . . . . . . . . . . . . 22
3.4 HPLC method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.5 Calibration of the polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.6 Determination of Tie Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.7 Partitioning of the Lysozyme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
19
In this chapter, the experimental methods to determine liquid-liquid phase equilibria, the binodal
curve and tie-lines of the studied ATPS and the lysozyme partitioning are presented. Firstly, the
materials used for the experiments are characterized.
3.1 Materials
3.1.1 Polymers
In this study PEG 6000, PEG 8000 and dextran were used as the linear polymers to form the
ATPS, while PFLDHB-G2-PEG6k-OH (G2) and PFLDHB-G3-PEG6k-OH (G3) were applied as the
branched polymers. All the chemicals were used without further purification. With the help of weak
heating and magnetic stirring, all polymers can be dissolved in water quickly.
The linear PEG polymers share the chemical formula HO(C2H4O)nH. PEG 6000, from Merck
Schuchardt OHG, Hohenbrunncan, Germany, can be characterized as a white or almost white solid
with a waxy or paraffin-like appearance. PEG 8000 is a white waxy powder from AMRESCO, 6681
Cochran Rd, United States.
The chemical formula of dextran is (C6H10O5)n and at room temperature it is a white power. It is
produced by Carl Roth GmbH + Co. KG, Karlsruhe.
All the mentioned polymers were stored at room temperature in a closed container opened only
for experiments.
The HBP both have the appearance of a white crystal and are produced by Polymer Factory
Sweden AB. The chemical structure of these polymers is presented in Figures 3.1 and 3.2.
The HBP were stored in an airtight container to protect from air humidity at 4°C, just opened for
the experiments.
Figure 3.1: Idealized structure of G2 [88].
20
Figure 3.2: Idealized structure of G3 [88].
In Table 3.1, the used polymers are further characterized.
Table 3.1: Characteristics of the used polymers.
Sample Molar mass MW (g/mol) Polydispersity MW /Mn Number of –OH per molecule
PEG 6000 5600-6600 – 2PEG 8000 7000-9000 – 2
Dextran 35000-45000 – –PFLDHB-G2-PEG6k-OH 6696 1,42 8PFLDHB-G3-PEG6k-OH 7625 1,3 16
3.1.2 Lysozyme
In this work, lysozyme from hen egg white, EC 3.2.1.17, purchased from SIGMA-ALDRICH CHEMIE
GmbH, Steinheim was used. The lysozyme has an average molar mass of 14600 g/mol and an activity
of 120 000 U/mg. The lysozyme is a lyophilized powder and it was stored at 4°C.
3.1.3 Solvents
For the experiments, deionised water was used as solvent to prepare all the solutions.
3.2 Determination of the binodal curves
Usually, there are three ways to determine the binodal curve for an ATPS, which are cloud-point
method, and turbidometric titration [89]. In turbidometric titration a series of systems are prepared
and titrated until a one-phase system is formed, the binodal lies just above this point. With the cloud
point method a concentrated stock of component 1, i.e., polymer X is added to a concentrated stock of
component 2, i.e., polymer Y. The solution is repeatedly taken above and below the cloud point. The
determination of nodes is accomplished by preparing a series of systems lying on different tie-lines
and analysing the concentration of components in the top and bottom phase. They all follow the same
21
principle: when the mixture is turbid, immiscibility happens; when the mixture is clear, it is still in the
homogeneous area.
In this work, the commonly used cloud point method was selected to determine the binodal curve.
Pure polymer aqueous solutions were prepared with a fixed compositions. The solution of one of the
polymers was added drop-wise into a solution of a different polymer with a known mass, by using a
syringe. The mixture was continuously stirred using a magnetic stirrer. After the addition of 3 ml, the
mixture was observed to determine if it turned turbid in the following 1-2 minutes. Since it did not turn
turbid, the last step was repeated until two phases formed, thus turning the mixture turbid. When the
point on the binodal curve was obtained, observed by the turbidity of the mixture, the added weight
was noted and 5 ml more were added. To obtain the second point of the binodal curve, deionized
water was added by the same method until the mixture turned clear. The two steps were repeated in
order to obtain the binodal points with a higher polymer B composition. The remaining binodal curve
can be determined by exchanging the polymers, i.e., adding the solution of polymer B to the solution
of polymer A. The experiment was conducted at room temperature.
The turbidometric titration was performed for the systems PEG 6000 – Dextran – Water, PEG 8000
– Dextran – Water and G2 – Dextran – Water. The weighing amount of polymers and water and wt%
can be seen in Appendix B. Results will be shown and discussed in Section 5.
3.3 Introduction to High Performance Liquid Chromatography
To determine the concentrations of linear branched polymers in ATPS, size exclusion chromatog-
raphy (SEC) equipped with refractive index detector (RID) was applied.
Solvent reservoir and filter
Pump to produce high pressure
HPLC tubeSample
injectionDetector
Processing unit and display
Waste
Signal to processorColumn oven
In-line degasser
Solvent and sample delivery system Separation system Detection system
Figure 3.3: Flow Sheet of HPLC based on [90].
Size Exclusion Chromatography (SEC) has been regularly used as a type of HPLC since 1959,
when Porathand Florian investigated a cross-linked dextran gel to separate proteins [90]. SEC is used
primarily for the analysis of large molecules such as proteins or polymers. SEC works by trapping
the smaller molecules in the pores of a particle, and allowing the larger ones to simply pass by the
22
pores as they are too large to enter them. Therefore, the larger molecules flow through the column
quicker than the smaller ones, meaning that the smaller the molecule, the longer the retention time. A
chromatogram is shown in Figure 3.4 as an example. The concentration of phase forming component
in the injected liquid can be determined by analysis of the areas of peaks with a data analysing tool,
as the peak area is proportional to its concentration.
0 1 0 2 0 3 0 4 0 5 0 6 0 7 0
0 , 0
0 , 5
1 , 0
1 , 5
2 , 0
RI (Vo
lts)
T i m e ( m i n u t e s )
Figure 3.4: Example of a typical chromatography diagram.
In this study, the column was tempered at 80°C with the help of an isothermal oven. The eluent,
deionised water, was filtrated with GHP membrane disc 47mm, 0,45µm, supplied by Pall Corporation,
USA, before being pumped into the HPLC.
The software and apparatus used are presented in Table 3.2.
Table 3.2: My caption
Software/Apparatus Model and Manufacturer
Intelligent Pump L-6200; Merck-Hitachi, Darmstadt and San Jose
Interface D-6000A; Merck-Hitachi, Darmstadt and San Jose
Refractive Index Detector (RID) K-2301; Dr. Ing. Herbert Knauer GmbH, Berlin
Chromatography Column SUPREMA analytical in Water; PPS Polymer Standards Service GmbH, Mainz
Intelligent Auto Sampler AS-4000; Merck-Hitachi, Darmstadt and San Jose
Isothermal Column Oven CTO-6AS; Shimadzu Deutschland GmbH, Duisburg
Software D-7000; Merck-Hitachi, Darmstadt and San Jose
The measured samples were prepared with Millipore water, the mentioned polymers and the pro-
tein lysozyme. Before measurement, the samples were diluted using with Millipore water. Before the
samples were injected into the vials, they were filtrated with a syringe filter of 25mm with 0,45mm
cellulose acetate membrane, purchased from VWR International, USA.
The RID measured the values used by the computer software to later create the chromatograms.
The chromatograms were analysed using the data analysis software OriginLab 9.0. To analyse the
23
peak areas and calculate the concentration of the solution, the built-in analysing tool named Peak
Analyzer was used.
3.4 HPLC method
Besides the mentioned separation by size, polymers are also prone to interact with surface charged
sites of the chromatographic stationary phase. These ionic interactions can result in the absorption of
the polymers, shifts in retention time or asymmetry. The mobile and stationary phase are selected in
order to avoid non-ideal interactions. [91] In this work, an ideal system for the study of polymers was
used. The mobile phase was deionised water and the stationary phase was a polyhydroxymethacry-
late copolymer network. No buffer was used.
Other factors can be used to manipulate SEC separations, such as flow rate, volume load and
temperature of the column. Adjustment of these factors can impact resolution, analysis time, and/or
sensitivity.
A high temperature operation has a low solvent viscosity and high diffusivity of the polymer
molecules [92]. In the used column, the analysed polymers have bordering retention times, so to
optimize the resolution the maximum column temperature, 80°C, was used.
Different flow rates were tested in order to optimize the chromatographic process. The selection
of the flow rate to use was made considering that the peaks of the analysed polymers need to have
retention times that ensure a good area of analysis. Although polymers are more effectively separated
at lower flow rates, the time consumption increases with decreasing flow rate. This should be avoided
because of the great number of measurements made in this work. In Table 3.3, the retention times
corresponding to different flow rates are presented, for the aqueous solutions of PEG 6000 and dex-
tran, as well as the calculated time differences. Millipore water was used as the eluent and washing
liquid for the auto sampler.
According to the results shown in Table 3.3, at 0,2 ml/min the two polymers can already be sepa-
rated with an appropriate retention time difference, as is shown in Figure 3.5. To reduce the measuring
time, a flow rate of 0,20 ml/min is used and the corresponding stop time is 62 minutes.
Table 3.3: Retention time differences between PEG 6000 and dextran for different flow rates at 80 °C.
flow rate (ml/min) retention time (min) Difference (min)
0,15 79,7 7,086,7
0,2 52,1 6,058,1
0,23 45,3 3,949,2
24
0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5 5 0 5 5 6 0 6 5- 0 , 1
0 , 0
0 , 1
0 , 2
0 , 3
0 , 4
RI (vo
lts)
T i m e ( m i n u t e s )
Figure 3.5: Chromatogram obtained from a PEG 6000 – Dextran aqueous solution at 80°C with a flow rate of0,2 ml/min.
In the chromatogram, the first peak corresponds to PEG 6000 and the second one to dextran.
For the measurements of the remaining used solutions (PEG 8000 – dextran, G2 – dextran and
G3 – dextran) the same method was used. Since the peaks showed a good resolution, as can be
observed in Figure 3.6, no other methods were tested to measure these polymers’ concentrations.
The calibration was confirmed by performing test measurements. Despite using the same method,
the measuring time had to be extended, as the retention time of these polymers is longer. In Table 3.4,
a summary of the method and measurement time is presented.
25
0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5 5 0 5 5 6 0 6 50 , 00 , 20 , 40 , 60 , 81 , 0
RI (vo
lts)
T i m e ( m i n u t e s )(a) PEG 8000 – dextran 40
0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5 5 0 5 5 6 0 6 5 7 0- 0 , 10 , 00 , 10 , 20 , 30 , 40 , 50 , 60 , 7
RI (vo
lts)
T i m e ( m i n u t e s )(b) G2 – dextran 40
0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5 5 0 5 5 6 0 6 5 7 00 , 00 , 20 , 40 , 60 , 81 , 01 , 21 , 41 , 6
RI (vo
lts)
T i m e ( m i n u t e s )(c) G3 – dextran 40
Figure 3.6: Chromatogram obtained from the studied aqueous solution at 80°C with a flow rate of 0,2 ml/min.
In Figure 3.6 it is possible to observe several peaks with retention times different from the peaks
used to analyse the polymers of the mixture, which are a consequence of the polydispersity of the
HBP and dextran. Most of these peaks can be ignored as they have very different retention times
comparing to the main peaks, except the peak with a retention time of approximately 67 min (seen
26
Table 3.4: Summary of the methods and measurement times.
Flow rate (ml/min) Temperature (°C) Measurement time (min)
PEG 6000 – Dextran 0,2 80 62PEG 8000 – Dextran 0,2 80 64
G2 – Dextran 0,2 80 70G3 – Dextran 0,2 80 70
in (b) and (c) of Figure 3.6), which has some interference with the HBP peak. However, as it also
appears on the calibration chromatograms without compromising the quality of the calibration curve,
it was considered not to interfere with the measurement.
3.5 Calibration of the polymers
It is possible to relate the HPLC measured peak area with the concentration of the component
responsible for this peak, and when the range of concentrations is small, the relationship between both
is considered linear. To relate the peak area with the components’ concentration, a calibration curve
was made for each of the used polymers. The calibration was performed for a concentration range
of 0,01 to 0,2 wt% for PEG 6000, PEG 8000 and the HBP G2 and G3. For dextran the concentration
range was of 0,01 to 0,3 wt%.
In this step, Millipore water was used as diluter, eluent and washing liquid for the auto sampler. All
the solutions were prepared at room temperature. The measurements were made in a balance scale
with an accuracy of 0.0001 g.
Bulk solutions were prepared for each polymer, with the higher concentrations shown in Table 3.5.
The remaining solutions were prepared by diluting the bulk solutions with water, and these are pre-
sented in Tables 3.6 to 3.10.
To determine the standard curves, the areas in the chromatography diagrams for each polymer
with different concentrations were analysed with the use of OriginLab 9.0.
Table 3.5: Prepared bulk solutions for calibration of the polymers.
Solution mpolymer mwater (wt %)
1 PEG 6000 0,1030 50,1103 0,20552 PEG 8000 0,1016 50,0834 0,20283 Dextran 0,1013 50,3368 0,20124 Dextran 0,0619 20,7532 0,29825 G2 0,1024 50,0047 0,20476 G3 0,1063 50,4473 0,2107
27
Table 3.7: Prepared PEG 8000 Samples for Calibration. Bulk solution 2 was used.
Solution Bulk solution (g) Water (g) (wt %)
12 11,8230 16,0122 0,151813 8,0039 16,3893 0,099114 3,9969 16,2836 0,049815 1,9957 16,4616 0,024616 0,7878 16,0002 0,0101
Table 3.6: Prepared PEG 6000 Samples for Calibration. Bulk solution 1 was used.
Solution Bulk solution (g) Water (g) (wt %)
7 11,7196 16,0232 0,15038 8,0130 15,9916 0,10309 4,0327 16,0443 0,051710 2,0164 16,0104 0,025911 0,7821 16,0113 0,0100
Table 3.8: Prepared Dextran Samples for Calibration. Bulk solution 3 was used.
Solution Bulk solution (g) Water (g) (wt %)
17 8,0194 16,0438 0,100618 3,9326 15,8570 0,049919 0,7852 15,9970 0,0099
Table 3.9: Prepared G2 Samples for Calibration. Bulk solution 5 was used.
Solution Bulk solution (g) Water (g) (wt %)
20 8,0200 16,0078 0,102621 4,0004 16,0896 0,050922 0,8737 16,1837 0,0111
Table 3.10: Prepared G3 Samples for Calibration. Bulk solution 6 was used.
Solution Bulk solution (g) Water (g) (wt %)
23 8,0646 16,9858 0,100024 4,0441 16,1970 0,052625 0,8237 16,0342 0,0108
3.6 Determination of Tie Lines
The tie lines are determined with the help of HPLC to measure the concentration of each compo-
nent in the separated phases.
Firstly, within the binodal curve, some ATPSs for each system were prepared. The prepared sys-
tems are described in Table 3.11. Then the prepared solutions were stirred with magnets stirring for
1-3 hours and put in a 25°C water bath for 72 hours to reach equilibrium – complete phase separation.
28
After 72 hours, the two phases were separated with a syringe. Considering the limited interval
of calibration, both the top and bottom phases were diluted with an appropriate amount of Millipore
water, to make sure that the concentration after dilution of each component was located within the
calibration interval. Thus, the concentration of each component in the two phases can be measured
with HPLC.
To control the quality of the tie lines, the Lever Rule was applied. When the de-mixing point of the
ATPS lies on the tie line, which was created by connecting two composition points for each phase,
experiments were performed with acceptable accuracy.
Table 3.11: Composition and measured values of the components of the systems used to obtain the tie lines.
System PEG (g) Dextran (g) Water (g) PEG (wt %) Dextran (wt %) Water (wt %)
PEG 60001 0,3381 0,6031 5,0966 5,5997 9,9887 84,41152 0,4581 0,6187 4,9861 7,5558 10,2047 82,23953 0,4775 0,6805 4,8509 7,9465 11,3249 80,7286
PEG 80001 0,3614 0,6689 6,9768 4,5135 8,3538 87,13272 0,5000 0,6851 6,837 6,2328 8,5402 85,22713 0,6064 0,7203 6,6904 7,5638 8,9845 83,4516
G2 1 0,6021 0,6124 4,8089 9,9960 10,1670 79,83702 0,4518 0,6269 4,9347 7,5132 10,4251 82,0617
G3 1 0,6014 0,607 5,1562 9,4491 9,5371 81,01372 0,4503 0,6184 4,9548 7,4757 10,2665 82,2578
3.7 Partitioning of the Lysozyme
The goal of this work is to study the partitioning of lysozyme in the chosen systems: PEG 6000 –
dextran – water, PEG 8000 – dextran – water, G2 – dextran – water and G3 – dextran – water.
Two tie-lines were analysed for each type of system. The systems were prepared with two
lysozyme concentrations, 0,3 and 1 wt %. These concentrations were chosen as they are below
the lysozyme solubility in water and in PEG 6000/8000 aqueous solutions.
The solubility in PEG 20000 solutions can be estimated by Boncina et al. correlations [93] for
a temperature of 25°C, 0,20M of NaCl and pH of 4 and 6, obtaining 24,2 and 2,1 wt %. As the
solubility of lysozyme in PEG solutions increases with the polymerization [93] and decreases with the
salt concentration [94][93], it is possible to assume that for solutions of PEG with molar masses of
6000 and 8000 and no dissolved salts, the lysozyme solubility is higher than 1 wt %.
Howard et al. [94] measured the solubility of lysozyme in aqueous solutions, in the presence of
NaCl. For a pH of 7, 25°C and 1,5 wt % of NaCl, the measured lysozyme solubility was 1,8 wt %.
As mentioned before, and has been proven by Howard et al., the solubility decreases with the salt
concentration, therefore the lysozyme concentration in water is higher than 1 wt %.
The solubility in dextran was not considered since no relevant information was found on the topic.
The lysozyme solubility in solutions with the used HBP is not known, and for that reason the same
concentrations were used.
29
In order to know the partitioning of the lysozyme, the concentration on the top and bottom phases
of the prepared systems must be measured. For this, different methods were attempted.
Firstly the HPLC was performed using the previous column. Since the column used is applicable
to neutral and anionic polymers from 100 to 30 000 000 Da, an applicability to the lysozyme measure-
ment was expected for pHs higher than the lysozyme’s isoelectric point (at which the lysozyme has
a negative charge). The column is stable in a pH range of 1,5 to 13 and the isoelectric point of the
lysozyme is 11,2 [95]. To choose the method’s pH, a range of 7 (lysozyme charge of +7) to 12 (charge
lower than -2) was tested. The tested lysozyme solutions were prepared with a concentration of 0,2
wt %. Mixtures with dextran and PEG 6000 were also tested. None of the attempts was successful.
Secondly, a silica column usually used for biomolecules analysis was tried, TSKgel G3000SWxl
purchased at Tosoh Bioscience GmbH, Stuttgart. The TSKgel column has a calibration range of
10000 to 500000 Da and operates at a pH of 2,5 to 7,5. The results were also not optimal.
For that reason the problem was approached using a different method: Spectrophotometry.
Spectrophotometry is a method to measure the light absorbance of a chemical substance, by
measuring the intensity of light as a beam of light passes through the sample solution. The basic
principle is that each compound absorbs or transmits light over a certain range of wavelengths [96].
A spectrophotometer is used to measure the amount of photons (the intensity of light) absorbed after
it passes through the sample solution. A spectrophotometer, in general, consists of two devices: a
spectrometer and a photometer. A spectrometer is a device that produces, typically disperses and
measures light. A photometer is a photoelectric detector that measures the intensity of light.
Knowing the intensity of the light before and after it passes through the sample, the transmittance,
T, can be calculated using Equation 3.1,
T =ItI0
(3.1)
where Io and It correspond to the light intensity before and after the beam of light passes through
the cuvette, respectively. Transmittance is related to the amount of photons that is absorbed, and is
designated as absorbance, A, by Equation 3.2.
A = −log(T )− log(ItI0
)(3.2)
Having the absorbance, the concentration of the component being measured can be determined
using Lambert-Beer Law, written as:
A = εlC (3.3)
where ε is the absorption coefficient, l is the path length, and c is the concentration.
Since proteins absorb light at a specific wavelength, a spectrophotometer can be used to directly
measure the concentration of a protein in solution. The number and type of amino acids of the protein
affect the absorption [96].
30
The typical Ultraviolet (UV) absorption spectrum of a protein can be divided in three distinct re-
gions: the far-UV region, below 210 nm; the region between 210 nm and 240 nm; and the near-UV
region, above 250 nm. In a protein, absorption in the far-UV region is caused by peptide bonds,
and in the near-UV region by the transfer of electron between the π-orbitals of aromatic amino acids.
The maximum absorption peaks centre around 295, 275 and 260 nm for tryptophan, tyrosine and
phenylalanine, respectively. In general, protein absorption of long-wavelength radiation is maximal at
around 280 [97].
The software and apparatus used are presented in Table 3.12.
Table 3.12: Information about apparatus and Software in spectrophotometry.
Software/Apparatus Model and Manufacturer
Spectrophotometer UVmini-1240 UV-Vis; Shimadzu Corporation,Kyoto Prefecture, Japan
Cuvette BRAND UV-Cuvette UV-Transparent Spectrophotometry; 11 Bokum Road12,5 x 12,5 x 45 mm
Software NEW UV DATA MANAGER; Shimadzu Corporation,Kyoto Prefecture, Japan
In this work the polymer absorption was also taken into account. In order to do that, the influence
of PEG and dextran in the spectrum of a lysozyme solution from 200 nm to 330 nm was evaluated.
Solutions with dextran concentrations ranging from 0 to 5 wt % where prepared (Table 3.13), and the
UV spectrum of each solution was measured. The solutions were prepared with Millipore water. A
initial aqueous solution of 0,2 wt % was prepared and mixed with a magnetic stirrer. The subsequent
solutions where obtained by adding a dextran solution of approximately 25 wt %, while mixing with
the magnetic stirrer. Four additions were made, obtaining the compositions presented in Table 3.13.
After every new addition a sample of 1 ml is taken from the mixture with a syringe, measured in the
spectrophotometer. The data was saved with the help of the optional NEW UV DATA MANAGER
software.
In Figure 3.7 the influence of dextran can be observed.
31
2 0 0 2 2 0 2 4 0 2 6 0 2 8 0 3 0 0 3 2 00 , 00 , 20 , 40 , 60 , 81 , 01 , 21 , 41 , 61 , 82 , 02 , 2
Abs
W a v e l e n g t h
Figure 3.7: Dextran influence in the lysozyme UV spectrum. The lighter the colour of the line, the higher thedextran concentration. The composition of the analysed solutions can be found in the Table 3.13.
Table 3.13: Composition of the solutions analysed in Figure 3.7.
Solution Lysozyme (wt %) Dextran (wt %)
0 0,0203 0,00001 0,0194 1,11712 0,0181 2,83463 0,0171 4,08674 0,0162 5,1962
Even though the lysozyme concentration decreases from 0,020 to 0,016 wt %, the absorbance in
the 280 nm wavelength increases with the dextran concentration. The same evaluation was made
for the PEG linear polymers, and, although less influential, similar results were observed. The ab-
sorbance variation can be a result of the absorption of the polymers, a conformational change of the
protein caused by the interactions with the polymer, or both.
The polymers’ influence precludes the measurement of the lysozyme concentration for the wave-
length of 280 nm, showed in Figure 3.7, the second maximum of the spectrum will be considered,
220 nm. To verify if the polymers have a negligible influence on the absorbance at 220 nm, measure-
ments of the lysozyme concentration were made with solutions of a known polymer concentration.
The measurements were also made for 280 nm, to compare the results.
A lysozyme stock solution was prepared with a concentration of 0,2 wt %, weighting the lysozyme
mass and dissolving it in Millipore water. The remaining solutions were prepared by diluting the stock
solution with Millipore water.
Firstly, the maximum lysozyme concentration for the applicability of the Lambert-Beer Law was
determined. The absorbance of the prepared solutions was measured and the values are presented
in Figure 3.8. Based on the data of Figure 3.8, the concentration range for 220 and 280 nm can be
defined as 0,001 to 0,01 wt % and 0,001 to 0,03 wt %, respectively.
32
0 , 0 0 0 , 0 2 0 , 0 4 0 , 0 6
0 , 0
0 , 5
1 , 0
1 , 5
2 , 0
Abs
L y s o z y m e ( % ( w / w ) )
Figure 3.8: Absorbance of lysozyme at 220 (squares) and 280 (triangles) nm.
Secondly, calibration curves where determined for lysozyme solutions with 1 and 7 wt % of dextran.
The solutions were prepared by adding a known amount of solid dextran to lysozyme solutions with a
known concentration.
The obtained points for the solutions with 1 and 7 wt % of dextran are presented in Figure 3.9. As
expected, for 280 nm the presence of dextran is noticeable by an increase in the interception value of
the line that describes the points’ tendency for the two dextran concentrations studied.
0 , 0 0 0 0 , 0 0 5 0 , 0 1 0 0 , 0 1 50 , 0
0 , 5
1 , 0
1 , 5
2 , 0
Abs
L y s o z y m e ( % ( w / w ) )
(a) Absorbance at 220 nm.
0 , 0 0 0 0 , 0 0 5 0 , 0 1 0 0 , 0 1 50 , 0
0 , 1
0 , 2
0 , 3
0 , 4
0 , 5
0 , 6
Abs
L y s o z y m e ( % ( w / w ) )
(b) Absorbance at 280 nm.
Figure 3.9: Influence of dextran concentration in the measurement of lysozyme concentration. Squarescorrespond to solutions with no dextran; circles and triangles correspond to solutions with 1 and 7 wt % dextran,
respectively.
The points obtained at 220 nm prove that, if the dextran concentration is lower than 1 %, the
measurement is not influenced by its presence. As the PEG polymers showed a lower influence on
the spectrum, the same assumption was made for these polymers.
Considering the results of Figure 3.9, the dilutions made before the measurement of lysozyme in
the bottom and top phases of the ATPS containing linear polymers guarantee combined concentra-
tions (dextran + PEG) lower than 1 wt %.
33
For the measuring of the HBP a different method was a attempted. The HBP absorbed UV light
at 220 nm. To account for this absorption a calibration curve for G2 was made, with which, knowing
the concentration of HBP in the systems, the absorbance contribution of these polymers can be
subtracted. The calibration was performed in a concentration range of 0,01 to 0,2 wt %.
A bulk solution was prepared by weighing a certain amount of polymer on a scale with an accuracy
of 0,0001 g and dissolving it with water. Other samples with lower concentration were diluted from
the bulk solution by adding certain amounts of water. Four solutions with different concentrations for
each polymer were prepared (as shown in Table 3.14).
Table 3.14: Prepared G2 samples for calibration in spectrophotometer.
Solution Dilution (g) G2 (mg) Water (g) (wt %)
Bulk solution 1 - 0,0113 5,122 0,22011 1,0147 of bulk sol. 1 2,2337 4,0227 0,04432 0,9927 of bulk sol. 1 2,1852 4,0366 0,00883 1,0024 of bulk sol. 1 2,2066 5,4997 0,00134 1,9099 of bulk sol. 1 4,2043 2,1788 0,1028
To test this method, the absorbance of two solutions with a known concentration of G2 and
lysozyme was measured, and the lysozyme concentration was calculated and compared with the real
one. The solutions were prepared by mixing a weighed amount of bulk solution 1 from Table 3.14, and
2 from Table 3.18 with Millipore water. The weighed amounts and final concentrations are presented
in Table 3.15.
Table 3.15: Prepared solutions to test the lysozyme measurement method in samples containing G2. BSstands for bulk solution.
Test solutions BS 1 (Table 3.18 (g) BS 2 (Table 3.14 (g) Water (g) Lysozyme wt % G2 wt %
1 0,2954 1,9927 1,9937 6,774E-03 4,785E-022 0,2017 2,9232 0 6,338E-03 4,148E-02
The systems were prepared with the same procedure as the one explained in section 3.6, with
the addition of lysozyme, which was measured and added to the mixture at the same time as the
polymers.
After mixing the systems with a magnetic stirrer and settling, the phases were separated as de-
scribed in section 3.6. ATPS consisting of PEG, Dextran, HBP, Millipore water and lysozyme were
prepared. The results in percentage of individual component are recorded in Tables 3.16 and 3.17.
Systems A correspond to a lysozyme concentration of 0,3 wt %, and systems B to a 1 wt % concen-
tration. System 1 corresponds to the shortest tie-line, nearer the critical point.
34
Table 3.16: Composition of prepared systems with lysozyme and linear polymers. The A correspond to systemswith 0,3 wt % of lysozyme and the B to systems with 1 wt %. System 1 is
System PEG (g) Dextran (g) Lysozyme (g) Water (g) Total mass (g) PEG (wt %) Dextran (wt %) Lysozyme (wt %) Water (wt %)
PEG 6000 (1)
A1 0,456 0,617 0,019 4,911 6,003 7,598 10,272 0,312 81,818A2 0,477 0,680 0,018 4,896 6,071 7,864 11,195 0,297 80,644B1 0,461 0,618 0,061 4,869 6,009 7,678 10,284 1,015 81,022B2 0,474 0,679 0,062 4,777 5,992 7,908 11,329 1,033 79,730
PEG 8000 (1)
A1 0,372 0,516 0,018 5,076 5,982 6,218 8,626 0,304 84,852A2 0,451 0,538 0,018 5,030 6,037 7,474 8,910 0,298 83,319B1 0,375 0,526 0,062 5,050 6,012 6,232 8,747 1,028 83,993B2 0,454 0,543 0,060 4,960 6,017 7,539 9,019 1,002 82,440
PEG 6000 (2)
A1 0,459 0,620 0,018 4,917 6,014 7,626 10,303 0,303 81,768A2 0,476 0,679 0,018 4,839 6,012 7,911 11,299 0,299 80,491B1 0,458 0,617 0,060 4,865 6,000 7,625 10,287 0,998 81,090B2 0,476 0,679 0,060 4,791 6,006 7,924 11,309 0,992 79,775
PEG 8000 (2)
A1 0,373 0,517 0,018 5,209 6,116 6,095 8,454 0,291 85,160A2 0,450 0,547 0,019 5,113 6,129 7,346 8,930 0,303 83,421B1 0,378 0,516 0,061 5,092 6,047 6,258 8,532 1,009 84,202B2 0,452 0,546 0,061 4,971 6,030 7,500 9,059 1,007 82,434
Table 3.17: Composition of prepared systems with lysozyme and HBP. The A correspond to systems with 0,3wt % of lysozyme and the B to systems with 1 wt %.
System HBP (g) Dextran (g) Lysozyme (g) Water (g) Total mass (g) HBP (wt %) Dextran (wt %) Lysozyme (wt %) Water (wt %)
G2
A1 0,720 0,730 0,018 4,546 6,014 11,968 12,138 0,306 75,588A2 0,453 0,624 0,019 4,919 6,014 7,532 10,370 0,319 81,779B1 0,722 0,724 0,061 4,511 6,017 11,990 12,035 1,012 74,962B2 0,451 0,618 0,061 4,885 6,015 7,499 10,274 1,009 81,218
G3 A1 0,721 0,721 0,018 5,270 6,731 10,715 10,715 0,273 78,296A2 0,455 0,621 0,018 4,916 6,009 7,568 10,327 0,298 81,807
For the linear polymers, duplicates were made for each system. For the HBP, no duplicates were
made. The systems with G3 and 1 wt % of lysozyme were also not made, as a result of the incapacity
to measure lysozyme concentration in this system, after partitioning.
To measure the lysozyme concentration in each phase, the solutions were diluted with Millipore
water with the dilution factors presented in Tables B.24 to B.27.
Before each composition is measured in the spectrophotometer, a calibration is essential. The
calibration was carried out with a concentration range of 0,001-0,01 wt % of lysozyme.
The calibration solutions were prepared from two bulk solutions, a first one with 0,2 wt % of
lysozyme and a second one with 0,1 wt %. The lysozyme was dissolved in water and the amount
of both was weighted on a scale with accuracy of 0,0001 g. Other samples with lower concentration
were diluted from the bulk solution by adding the amounts of water indicated on Table 3.18. The two
bulk solutions were prepared on different days, and the measurement of respecting diluted solutions
also, confirming the applicability of the calibration. Nine solutions with different concentrations were
prepared (as shown inTable 3.18).
35
Table 3.18: Prepared lysozyme samples for calibration in spectrophotometer.
Solution Dilution (g) Lysozyme (mg) Water (g) (w- %)
Bulk solution 1 - 0,0503 24,9981 0,20081 0,5410 of bulk sol. 1 1,0864 10,1215 0,01072 0,2889 of bulk sol. 1 0,5801 10,1344 0,00573 0,1222 of bulk sol. 1 0,2454 10,0091 0,00254 0,0656 of bulk sol. 1 0,1317 10,3680 0,00135 0,4118 of bulk sol. 1 0,8269 10,3666 0,0080
Bulk solution 2 - 0,0499 50,7701 0,09826 1,0419 of bulk sol. 2 1,0230 9,9630 0,01037 0,7987 of bulk sol. 2 0,7842 10,0888 0,00788 0,5305 of bulk sol. 2 0,5209 12,5136 0,00429 0,2006 of bulk sol. 2 0,1970 19,4468 0,0010
36
4Model
Contents4.1 Lattice Cluster Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.2 Wertheim theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
37
4.1 Lattice Cluster Theory
In this chapter the model used, a version of the LCT applicable for a multicomponent polymer
solution/blend, will be introduced.
Freed and co-workers [25–29] improved the FH theory by identifying and understanding its de-
ficiencies, and calculating corrections. On the basis of de Gennes’ work [98] treating long chain
molecules as a self-avoiding walk on a lattice by the use of spins they developed the LCT for long
chain molecules with different architectures. The main amendments will now be addressed.
As mentioned in Section 2.5.4.A, the FH lattice model treats the monomers as structureless units.
The LCT overcomes this limitation by allowing monomers to occupy several neighbouring lattice sites
to mimic the sizes, shapes and structure of the actual molecules, as can be seen in Figure 4.1 [29].
PIBPEPPPPE
Figure 4.1: United atom group models for monomers of poly(ethylene) (PE), poly(propylene) (PP), poly(thylenepropylene) (PEP) and poly(isobutylene). Circles represent CHn groups, solid lines designate C - C bonds inside
the monomer, and dotted lines indicate C - C bonds linking the monomer to its neighbours along the chain.
The non-combinatorial contribution of the model is generated analytically and accounts for the
compositions, molecular weights, nearest neighbours, attractive van der Walls interactions, and tem-
perature. This way, the use of the purely empirical FH parameter is avoided.
In this study, a model from Kulaguin and Zeiner [13] is used.
The LCT derives in the form of a cluster expansion in the inverse coordination number 1/z and the
reduced interaction energy ∆εij/kBT . Using Tables I-III [26] and the corrections given in [27], Equa-
tion (4.1) is defined for the Gibbs free energy contribution of the LCT, GLCT , for an incompressible
LCT with multi-components.
GLCTNLRT
=
n∑i=1
ΦiMi
ln (Φi)−∆S
NLRT− ∆E1
NLRT− ∆E2
NLRT(4.1)
The first term of Equation (4.1) emerges as the contribution of the classical FH entropy, ∆S is the
non-combinatorial correction term of the entropy, and ∆E1 and ∆E2 are the energy contributions of
first and second order.
The correction term of the entropy and the energy contributions are calculated with the use of
Equations (4.2)-(4.4).
∆SNLRT
= 1z
n∑k=1
n∑λ=1
XS1,kλΦkΦλ− 2
z2
n∑k=1
n∑λ=1
XS2,kλΦkΦλ + 1
z2
n∑k=1
n∑λ=1
XS3,kλΦkΦλ
+ 4z2
n∑k=1
n∑λ=1
n∑µ=1
XS4,kλµΦkΦλΦµ + 2
z2
n∑k=1
n∑λ=1
n∑µ=1
n∑η=1
XS5,kλµηΦkΦλΦµΦη
(4.2)
38
∆E1
NLRT= 1
z
n∑k=1
n∑λ=1
n∑µ=1
Xε1,kλµ∆εkλΦkΦλΦµ + 1
z
n∑k=1
n∑λ=1
Xε2,kλΦkΦλ
n∑µ=1
∆εkµΦµ
+ 1z
n∑k=1
n∑λ=1
Xε3,kλΦkΦλ
n∑µ=1
n∑η=1
∆εµηΦµΦη + 1z
n∑k=1
n∑λ=1
Xε4,kλΦkΦλ
n∑µ=1
∆ελµΦµ
− 2z
n∑k=1
n∑λ=1
n∑µ=1
Xε5,kλµΦkΦλΦµ
n∑τ=1
n∑η=1
(∆ετη − 2∆εkτ ) ΦτΦη
− 12
n∑k=1
Xε6,kΦk
n∑µ=1
n∑η=1
(∆εµη − 2∆εkµ) ΦµΦη − z4
n∑µ=1
n∑η=1
∆εµηΦµΦη
(4.3)
∆E2
NLRT=
n∑k=1
Xε2
1,kΦkn∑µ=1
∆ε2kµ4 Φµ +
n∑k=1
n∑λ=1
Xε2
2,kλΦkΦλn∑µ=1
n∑η=1
∆εkµ∆εµη2 ΦµΦη
+n∑k=1
n∑λ=1
Xε2
3,kλΦkΦλn∑µ=1
n∑η=1
∆εkµ∆εkη4 ΦµΦη
+k∑k=1
k∑λ=1
Xε2
4,kλΦkΦλk∑µ=1
k∑η=1
k∑ω=1
k∑τ=1
∆εωτ∆εµη4 ΦµΦηΦωΦτ
+k∑k=1
k∑λ=1
Xε2
5,kλΦkΦλk∑µ=1
k∑η=1
k∑ω=1
∆εµη∆εηω4 ΦµΦηΦω
+k∑k=1
k∑λ=1
Xε2
6,kλΦkΦλk∑µ=1
k∑η=1
k∑ω=1
∆εkµ∆εηω2 ΦµΦηΦω
+k∑k=1
k∑λ=1
Xε2
7,kλΦkΦλk∑µ=1
k∑η=1
(∆ε2kλ
4 −∆εkµ∆εkλ + ∆εkµ∆ελη − ∆εkµ∆ελµ2 +
∆εkλ∆εµη2
)ΦµΦη +
k∑k=1
Xε2
8,kΦkk∑µ=1
k∑η=1
∆ε2µη8 ΦµΦη
+ z4
k∑µ=1
k∑η=1
k∑ω=1
k∑τ=1
(∆εµη∆εωτ
4 − ∆εµη∆εηω2 +
∆ε2µη4
)ΦµΦηΦωΦτ
(4.4)
The calculated corrections are implemented in the model used, as the terms XSi , Xε
i and Xε2
i ,
defined by Equations eqs. (A.1) to (A.5), eqs. (A.6) to (A.11), and eqs. (A.12) to (A.18) [13], that can
be found in Appendix A, including a temperature-independent contribution to the χ parameter that
arises from the corrections introduced by the chain connectivity and excludes volume constraints.
In the X terms, two structural parameters, b3,i and b4,i – the number of branching points of three
and four – are introduced in order to account for the chains’ architecture. The use of this structural
parameters assumes that there are at least two segments between two branching points, which for
the used hyperbranched polymer this assumption is fulfilled. To consider the short chain branching
the equations given in [99] have to be used. The advantage of the equations of this work is the simple
way of describing the architecture and the reduced number of parameters.
In the framework of LCT, the interaction parameter ∆εij is related to one contact point between
segments. Hence, the product of coordination number and interaction energy describes the interac-
tion of one segment. When the coordination number z tends to infinity (z →∞), Equations (4.2)-(4.4)
tend to zero (∆E1,∆E2,∆S → 0), i.e., the entropic LCT-contribution is reduced to the classical FH
entropy and, in case the interaction energy also tends to zero (∆εij → 0), the energy contributions
are reduced to the FH χ−parameter.
39
4.2 Wertheim theory
The LCT presented in the previous chapter 4.1 has a disadvantage that cannot be ignored in the
systems to be studied in this work, which is the inability to describe the molecules’ association. This
is particularly important as the Dextran T40 molecules have 3 OH-groups per glucose unit and the
PEG/HB has the association sites on the oxygen.
The association between molecules in the mixtures is usually described based on one of two
approaches. The ”chemical theory” assumes the formation of hydrogen bonds as a chemical reaction,
and the ”physical theory” calculates the association contributions using integral equations consisting
of an interaction potential to illustrate the association interactions [100].
Wertheim [30, 31] developed one ”physical theory” that treats the associative bonds as directional
forces between two association sites. This theory takes into account self and cross associations, i.e.,
between molecules of the same type and different types. The Wertheim theory was transferred on
a fully occupied lattice, so an incompressible fluid is regarded. As a result of the application of the
Wertheim theory, the Gibbs energy has the extra addition (4.5)[101].
GassoNLRT
=∑i
Φi
[∑Ai
[ln (XAi)−
XAi
2
]+
1
2Nassoi
](4.5)
Where Nassoi is the number of association sites per segments of component i, and XAi are the
non-bonded segment fractions.
To apply this theory it is necessary to identify every associative site in the studied molecules [30,
31], which are divided by class according to the type of group that is responsible for the association,
such as OH, amine or carbonyl. The non-bonded segment fractions can be calculated with Equation
(4.6) [102][101]
XAi =
1 +∑j
∑Bj
ΦjXBj∆ij
−1
(4.6)
The cross association strength between two molecules ∆ij is defined as:
∆ij = Kassoij
[exp
(εassoij
kBT
)− 1
](4.7)
where Kassoij is the association volume and εassoij is the association energy between two segments
of two different components. These cross association parameters are calculated based on the self
association ones, that must be fitted to experimental data:
Kassoij =
Kassoii +Kasso
jj
2(4.8)
εassoij = (1− kij)√εassoii εassojj (4.9)
where the parameter kij is introduced to consider the deviation of the association energy from
the geometrical mixing rule, i.e., the difference between√εassoii εassojj and εassoij ; Kasso
ii or Kassojj , and
40
εassoii or εassojj are the association volumes and the association energies of one single component,
respectively.
Finally, the total Gibbs free energy of the systems, when accounting for the architecture and the
association interaction, is defined as:
GmixNLRT
=GassoNLRT
+GLCTNLRT
(4.10)
41
5Results and Discussion
Contents5.1 Liquid-liquid equilibria for the lysozyme-water system . . . . . . . . . . . . . . . 43
5.2 Liquid-liquid equilibria for ternary systems . . . . . . . . . . . . . . . . . . . . . . 44
5.3 Partitioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
42
In this chapter, solutions of two linear PEG polymers and two Hyperbranched Polymers (HBP) are
studied. To describe the architecture of the HBP, the number of branching points of two of these, and
the number of segments will be used as structural parameters. Moreover, the partitioning of lysozyme
will be studied in the linear system and modelled with a quaternary LCT combined with the Wertheim
theory.
5.1 Liquid-liquid equilibria for the lysozyme-water system
Figure 5.1: Lysozyme structure illustration [103]
In the following chapter, the parameters describing the LLE of the binary subsystem lysozyme-
water will be adjusted to experimental data with the LCT in combination with the Wertheim theory.
This step will facilitate the parameter adjustment in the quaternary system, where the partitioning of
the lysozyme will be calculated.
For the modelling of the LLE of lysozyme solution, a linear structure will be considered and the
3D configuration will not be taken into account. The number of segments was defined based on the
amino acids of the protein, so it equals 147, and the branching parameters are zero. The binary
interaction, association and LCT parameters were fitted to the experimental results.
The LLE experimental data, taken from Taratuta et al [104], and the modelled data are shown in
Figure 5.2.
43
0 , 0 0 0 , 0 5 0 , 1 0 0 , 1 5 0 , 2 0 0 , 2 5 0 , 3 0 0 , 3 5 0 , 4 0 0 , 4 5 0 , 5 0 0 , 5 52 6 4
2 6 6
2 6 8
2 7 0
2 7 2
2 7 4
2 7 6
2 7 8
2 8 0
Temp
eratur
e (K)
C o n c e n t r a t i o n ( % ( w / w )
Figure 5.2: Liquid-liquid equilibrium of aqueous lysozyme solution. Symbols are experimental data []; Thebroken line is the model. Experimental data taken from Taratuta et al [104], from solutions in phosphate buffer
with a pH of 7.
Table 5.1: Binary interaction, association and LCT parameters for the aqueous lysozyme solution.
Binary interaction energy (∆εijkB
) [K] -25
Association energy ( εassoij
kB) [K] 559
Association parameter (Kassoij ) 0,1
Deviation of the geometrical mixing rule (kij) 0,1
5.2 Liquid-liquid equilibria for ternary systems
In this chapter, the ternary systems of water/dextran-T40 and PEG 6000, 8000, G2 or G3 are
studied. The experimental and modelled results will be shown.
5.2.1 Calibration for tie-line determination
After selection of the appropriate HPLC method, the calibration lines were investigated according
to the procedure introduced in section 3.4. Results are displayed in x-y diagram with the x-axis
standing for the weight percentage of prepared polymer solutions, and the y-axis standing for the
area values obtained from the OriginLab evaluation. In the calibration diagrams, the relations between
evaluated areas and corresponding concentrations are presented with filled dots. Calibration results
of linear polymers and HBP are displayed in Figure 5.3 and Figure 5.4, respectively.
44
0 , 0 0 0 , 0 5 0 , 1 0 0 , 1 5 0 , 2 0 0 , 2 50
1
2
3
4
5
6Are
a
C o n c e n t r a t i o n w t %
(a) PEG 6000a= -0,32±0,05; b= 28,9±0,4
R2=0,99795
0 , 0 0 0 , 0 5 0 , 1 0 0 , 1 5 0 , 2 0 0 , 2 50
1
2
3
4
5
6
Area
C o n c e n t r a t i o n w t %
(b) PEG 8000a= -0,3759±0,03; b= 28,8±0,3
R2=0,99858
0 , 0 0 0 , 0 5 0 , 1 0 0 , 1 5 0 , 2 0 0 , 2 5 0 , 3 0 0 , 3 50123456789
1 0
Area
C o n c e n t r a t i o n w t %
(c) Dextrana= 0,03±0,01; b= 29,06±0,09
R2=0,99982
Figure 5.3: Calibration Diagram for the linear polymers. The data is presented in the Table B.1. Therepresented calibration curve is defined by y = ax+ b.
0 , 0 0 0 , 0 5 0 , 1 0 0 , 1 5 0 , 2 0 0 , 2 50
1
2
3
4
5
6
Area
C o n c e n t r a t i o n w t %
(a) G2a= -0,07±0,05; b= 17,0±0,4
R2=0,99564
0 , 0 0 0 , 0 5 0 , 1 0 0 , 1 5 0 , 2 0 0 , 2 50
1
2
3
4
5
6Are
a
C o n c e n t r a t i o n w t %
(b) G3a= -0,06±0,04; b= 18,8±0,4
R2=0,99743
Figure 5.4: Calibration Diagram for the HBP. The data is presented in the Table B.2. The representedcalibration curve is defined by y = ax+ b.
All the calibration curves have a good R-square. The curves do not have an interception value of
zero, which can be explained by the low sensibility of the HPLC method to these polymers, probably
caused by their polydispersity. The polydispersity causes the different molar size polymer segments
to elute the column in various retention times, and consequently, not all the polymer is accounted for
in the measured peak. This effect is more noticeable in the PEG polymers, for which the samples with
the lower concentration (0,01 wt % in Table B.1) result in an almost null measured area.
5.2.2 Tie-line modelling
With the help of the Programming Software Free Pascal V2.6.4, the LLE will be calculated. The
LCT+Wertheim model, which has been introduced in the theoretical part, has already been imple-
mented to the program by Laboratory of Fluid Separations, TU Dortmund. Therefore, no program-
45
ming work is required any more. The only work that needs to be done is calculating with this already
existing program.
Firstly, the systems with linear polymers (PEG 6000/Dextran 40 and PEG 8000/Dextran 40) are
calculated maintaining the LCT and Wertheim parameters coherent. Secondly, a similar work will be
done for the systems with HBP, taking into account extra structural parameters this time.
The association parameters, association energy ( εassoii
kB) and association volume (Kasso
ii ) of the pure
solvent and the linear polymers were the first to be defined, based on former articles [13], and are
shown in Table 5.2
Table 5.2: Association parameters of water and linear polymers taken from [13].
Association energy PEG 6000 750
Association energy PEG 8000 750
Association energy Dextran 869
Association energy water 1512
Association volume PEG 6000 0,009
Association volume PEG 8000 0,009
Association volume Dextran 0,012
Association volume water 0,01
In the calculations for the linear polymer systems, the structural parameters taken into account are
the molar mass of each component, MM i, and the corresponding number of segments, Mi. In this
case, give the linear structure, the number of branching points of 3, b3,i, and 4, b4,i is zero. For the
PEG polymer the number of segments is calculated by setting the segment size as the molar mass of
the solvent, in this case water, meaning that each segment has a molar mass of 18 g/mol. Then, the
number of segments of each polymer is calculated as shown in Equation (5.1).
Mi =MM i
18(5.1)
The dextran molecule has rings as part of its structure, which cannot be described by the LCT.
Considering that the space occupied by the glucose rings cannot be disregarded, for each ring an
extra segment is accounted for. Thus, the number of segments per glucose is set as 7 – the number
of carbons plus one.
The interaction energies show a dependence on molar mass that was described for PEG, dextran
and water by Kulaguin Chicaroux and Zeiner [105] by Equations (5.2) to (5.4).
εPEG−water
kB= −7, 143× 10−5K ·mol
g·MM,PEG − 4, 771 K (5.2)
εDex−waterkB
= −5, 727× 10−5K · gmol
·MM,Dex − 5, 229 K (5.3)
εPEG−Dex
kB= 8, 166× 10−5 ·MM,Dex + 9, 884 K (5.4)
46
The dependence shown was used for an initial calculation of these parameters, followed by an
adjustment to the experimental data. In Table 5.3 the parameters calculated with the Equations (5.2)
to (5.4) are presented next with the adjusted ones.
Table 5.3: Interaction parameters of LCT+Wertheim for systems with linear polymers.. Calculated byEquations (5.2) to (5.4) and adjusted to the experimental results.
Calculated Adjusted Difference (%)
PEG 6000 - water -5,2 -5,4 4PEG 8000 - water -5,3 -4,8 11
Dextran - water -7,3 -9,5 23Dextran - PEG 6000 12,9 14,5 11Dextran - PEG 8000 12,9 13,5 4
The adjusted parameter that shows a bigger difference with the calculated is the value that de-
scribes the interaction between dextran and water. This can be related with the polydispersity of the
dextran (35000-45000), which is not taken into account by the model.
In addition to the molecular structure parameters, the coordination number z also needs to be
fixed. It describes the number of neighbour segments or next segments in the liquid. Originally, the
LCT was developed at z=6, as a cubic lattice [25]. In this work, the coordination is also considered as
6.
The modelled ATPS and the correspondent experimental data are illustrated in Figure 5.5, and the
used values can be found in Table 5.4.
0 5 1 0 1 5 2 0 2 5 3 00
5
1 0
1 5
2 0
2 5
3 07 0
7 5
8 0
8 5
9 0
9 5
1 0 0
D e x t r a nW a t er
P E G 6 0 0 0
(a) ATPS consisting of PEG 6000 – dextran 40 – water.
0 5 1 0 1 5 2 0 2 5 3 0
7 0
7 5
8 0
8 5
9 0
9 5
1 0 0 0
5
1 0
1 5
2 0
2 5
3 0
D e x t r a nW a t er
P E G 8 0 0 0
(b) ATPS consisting of PEG 8000 – dextran 40 – water.
Figure 5.5: Experimental and LCT+Wertheim points of ATPS consisting of linear polymers, in wt % at 298.15K:Symbols are experimental data []; The broken lines are the measured tie lines; the full lines are the modelled
tie lines and binodal curve. The parameters used are presented in ??
Comparing the modelling results and the experimental data, it can be stated that the binodal curve
can be described in good accordance with experimental data, but there are some deviations of the
47
Table5.4:
Sum
mary
ofLCT
andW
ertheimA
ssociationparam
etersetforlinearpolymers.
Com
ponentpropertiesS
tructuralparameters
Interactionparam
eters(∆εij
kB
)A
ssociationparam
eters
Association
Energy
Association
Volume
MMPEG
6000
6000MPEG
6000
333P
EG
6000-Dextran
14,5P
EG
6000750
PE
G6000
0,009
MMPEG
8000
8000MPEG
8000
444P
EG
8000-Dextran
13,5P
EG
8000750
PE
G8000
0,009
MMDextra
n37000
MDextran
1597D
extran-water
-9.5D
extran869
Dextran
0,012
PE
G6000-w
ater-5,4
Water
1512W
ater0.01
PE
G8000-w
ater-4,8
kasso
ijP
EG
6000-Water
0
kasso
ijP
EG
8000-Water
0
kasso
ijP
EG
6000-Dextran
0,045
kasso
ijP
EG
8000-Dextran
0,045
kasso
ijD
extran-water
0
48
calculated tie lines and the experimental ones. The bigger deviations correspond to the tie-lines near
the critical point, which are obtain with a bigger error experimentally, give that the equilibrium is more
difficult to achieve. Moreover, the tie-lines near the critical point were only measured once, making
the results less reliable than the other two.
Although there are deviations in the tie lines, the agreement between experimental and modelling
data is good for the linear system. Therefore, this LCT-Wertheim model can be applied to describe
liquid-liquid equilibrium of PEG6000-Dextran-water and PEG8000-Dextran-water system, and it will
be used to calculate the lysozyme partitioning in these systems.
In contrast to the simple architecture of the linear polymers, the HBP architecture must be specified
in a more complex manner in the calculation. The architecture will not be considered an adjustable
parameter, but rather it will be determined by the chemical structure of the molecules, similar to
former articles [13]. Firstly, the polymers’ architecture will be inspected in order to define the number
of branching points of 3, b3, and 4, b3, for each HBP.
The first considered HBP is G2, and its idealized structure can be seen in Figure 3.1. This polymer
is a hyperbranched polyester of pseudo generation 2, i.e., with one branching level in one of the ends.
It is composed by a core of PEG with a theoretical molar mass of 6000 and its branching units are
2,2-bis(methylol)propionic acid (bis-MPA).
The number of segments of the HBP were defined considering that each bis-MPA molecule, with
a molar mass of 133, occupies 5 segments of the lattice, thus defining that each segment has a molar
mass of 27:
Segment size =MM,MPA
5(5.5)
therefore the G2 molecule has a total of 248 segments. Regarding the architecture, this HBP has
only one branching point of 3, thus b3 = 1 and b4 = 0.
The G3 is represented in Figure 3.2 and it is also a hyperbranched polyester composed by a core
of PEG with a theoretical molar mass of 6000 and branching units of bis-MPA. This HBP is of pseudo
generation of 3, with 2 branching point levels. Once again, the segment size is 27, by the same
consideration, therefore G3 has a total of 346 segments. As this HBP has 3 branching points of 3,
b3 = 3.
The interaction and association parameters where adjusted to the experimental data.
The calculated and experimental values for both polymers are presented in Figure 5.6 and the
used adjusted parameters can be found in Table 5.5.
49
0 5 1 0 1 5 2 0 2 5 3 0 3 50
5
1 0
1 5
2 0
2 5
3 0
3 56 5
7 0
7 5
8 0
8 5
9 0
9 5
1 0 0
D e x t r a nW a t er
P F L D H B - G 2 - O H
(a) ATPS consisting of G2 – dextran 40 – water.
0 5 1 0 1 5 2 0 2 5 3 00
5
1 0
1 5
2 0
2 5
3 07 0
7 5
8 0
8 5
9 0
9 5
1 0 0
D e x t r a nW a t er
P F L D H B - G 3 - O H
(b) ATPS consisting of G3 – dextran 40 – water.
Figure 5.6: Experimental and LCT+Wertheim points of ATPS consisting of HBP, in wt %: Symbols areexperimental data []; The broken lines are the measured tie lines; the full lines are the modelled tie lines and
binodal curve. The used parameters are presented in Table 5.5
Comparing the modelling results and the experimental data, it can be stated that, for the ATPS
formed by G2 polymer, the binodal curve can be described in good accordance with experimental
data, but there are big deviations between the calculated tie lines and the experimental ones. This
can be explained by the chosen structural parameters, as these are the very influential parameters
in the tie-line slope. Although defining the bis-MPA monomer as occupying five segments in the
lattice allows for good tie-line calculations for the G3 system, as can be seen in Figure 5.6b, this
consideration did not apply for the G2 system. One reason for this could be the polydispersity of the
HBP and the dextran. Consideration of dispersity can be an option to optimize the model.
Similar systems, as PEG4000-dextran110, PEG10000-dextran40 and PEG10000-dextran110, can
also be modelled with LCT and Wertheim theory [105]. Although deviations appear in some systems,
this model can be accepted to describe the LLE of ATPS consisting of two polymers.
Additionally, a comparison can be made between the linear and the hyperbranched systems. Con-
sidering the molar mass similarity, in Figure 5.7 the PEG 6000 system is compared with G2 and the
PEG 8000 with G3. Comparing the ATPS based on the HBP and the ATPS including the PEG 6000
and 8000, it can be seen that the ATPS containing linear polymers has steeper tie lines. One disad-
vantage of the ATPS based on HBP is that the weight fraction of HBP polymer of the top (HBP-rich)
phase is higher than in the ATPS containing linear polymers, which results in a higher requirement of
HBP to form an ATPS. This fact is confirmed by the translation of the critical point in the ternary sys-
tem to the right, meaning the minimum required amount of polymers to form an ATPS in both cases,
(a) and (b) Figure 5.7, is larger for the systems containing HBP.
50
0 5 1 0 1 5 2 0 2 5 3 00
5
1 0
1 5
2 0
2 5
3 07 0
7 5
8 0
8 5
9 0
9 5
1 0 0
D e x t r a nW a t er
P E G 6 0 0 0 / G 2
(a) G2 vs PEG 6000.
0 5 1 0 1 5 2 0 2 5 3 00
5
1 0
1 5
2 0
2 5
3 07 0
7 5
8 0
8 5
9 0
9 5
1 0 0
D e x t r a nW a t er
P E G 8 0 0 0 / G 3
(b) G3 vs PEG 8000.
Figure 5.7: Comparison can be made between the linear and the hyperbranched systems, in wt % at 298.15K:full line corresponds to the HBP systems; broken lines correspond to the linear polymers’ systems.
51
Table5.5:
Sum
mary
ofLCT
andW
ertheimA
ssociationparam
etersetfortheH
BP
.
Com
ponentpropertiesS
tructuralparameters
Interactionparam
eters(∆εij
kB
)A
ssociationparam
eters
Association
Energy
Association
Volume
MMG
26696
MG
2248
G2-D
extran20
G2
785G
20,0055
MMG
37625
MG
3346
G2-W
ater-5
G3
779G
30,009
G3-D
extran15
kasso
ijG
2-Water
0
G3-W
ater-6
kasso
ijG
3-Water
0
kasso
ijG
2-Dextran
0,085
kasso
ijG
3-Dextran
0,045
52
5.3 Partitioning
In this section, the partitioning of the lysozyme in the ATPS with the linear polymers PEG 6000,
8000 and dextran will be studied. The partition was measured experimentally and modelled with a
quaternary LCT model in combination with the Wertheim theory.
5.3.1 Calibration for lysozyme quantification
A calibration was necessary before the partitioning coefficients could be determined. Results are
displayed in an x-y diagram with the x-axis standing for the weight percent of the prepared lysozyme
solutions while the y-axis stands for the the absorbance of the solution. Results of calibration for
partitioning investigation are illustrated in Figure 5.8.
0 , 0 0 0 0 , 0 0 4 0 , 0 0 8 0 , 0 1 20 , 0
0 , 5
1 , 0
1 , 5
2 , 0
Abs
L y s o z y m e w t %
Figure 5.8: Calibration diagram for the lysozyme. The data is presented in the Table B.3. The representedcalibration curve is defined by y = (144±3)x + (0,04±0,02). R2=0,99654.
In order to account for the HBP absorbance a calibration curve was made for the G2 molecule.
Results are displayed in an x-y diagram with the x-axis standing for the weight percent of the prepared
G2 solutions while the y-axis stands for the the absorbance of the solution. Results of the calibration
are illustrated in Figure 5.9.
53
0 , 0 0 0 , 0 2 0 , 0 4 0 , 0 6 0 , 0 8 0 , 1 0 0 , 1 20 , 0
0 , 5
1 , 0
1 , 5
Abs
G 2 w t %
Figure 5.9: Calibration diagram for G2 spectrometry. The data is presented in the Table B.3. The representedcalibration curve is defined by y = (11,9±0,2)x + (0,025±0,009). R2=0,99943.
To test the applicability of the presented calibration curve, two test solutions were prepared with
previously known lysozyme and G2 concentrations. Knowing the G2 concentration, CG2, the corre-
sponding absorbance, Acalc, was calculated with the calibration curve, and subtracted to the mea-
sured absorbance of the prepared solutions solutions, Ameas, obtaining an estimated absorbance,
Aest.
Aest = Ameas −Acalc (5.6)
With the estimated absorbance and the lysozyme calibration curve presented in Figure 5.8, the
lysozyme concentration was calculated and compared with the real one. In Table 5.6 the calculated
values are presented. To avaluate the applicability of the method, an error was calculated as:
Error (%) =|CLys,est − CLys,exp|
CLys,exp× 100 (5.7)
Where CLys,est is the estimated concentration of lysozyme and CLys,exp is the known experimental
concentration.
Table 5.6: Prepared and calculated concentrations, measured and calculated absorbances, and error fo thesolutions used on the explained test.
Experimental Estimated
Test solutions Lysozyme wt % G2 wt % Acalc Ameas Aest Lysozyme wt % Error (%)
1 6,774E-03 4,785E-02 0,571 1,670 1,099 7,29E-03 7,62 6,338E-03 4,148E-02 0,495 1,420 0,925 6,08E-03 4,1
Analysing the results, it can be considered that the lysozyme concentration can be estimated with
less than 10 % of error. This error can be related with the cumulative experimental error throughout
the procedure.
54
5.3.2 Modelling
The lysozyme partitioning was investigated by preparing systems with mixing points previously
studied and adding the lysozyme in two different concentrations, 0,3 and 1 wt % (designated A and B,
respectively) in order to evaluate the influence of the lysozyme concentration in the partitioning. For
each type of system, PEG 6000 – dextran, PEG 8000 – dextran, G2 – dextran and G3 – dextran, two
mixing points far from the critical point were chosen. This was defined in order to study the influence
of the TLL in the partitioning. The compositions of the prepared systems are presented in Tables
3.16 and 3.17, and the measured values can be found in Appendix B. Systems A correspond to a
lysozyme concentration of 0,3 wt %, systems B to a 1 wt % concentration. Systems 1 correspond to
the shortest tie-line, nearest to the critical point.
After mixing and settling, two transparent phases were obtained. In the systems with 1% of
lysozyme a white deposit was detected in the interface, resulting of lysozyme precipitation.
To ensure the values of the experimental points, each system was prepared twice, but the study
of the results was made with the average values of partitioning obtained for each system, presented
in Table 5.7.
Table 5.7: Average mixing point of the partitioning experiments.
PEG (%-w) Dextran (%-w) Lysozyme (%-w) Water (%-w)
PEG 6000 (1) 7,612 10,288 0,307 82,100PEG 6000 (2) 7,903 11,275 0,299 80,523PEG 8000 (1) 6,157 8,585 0,298 84,960PEG 8000 (2) 7,410 8,920 0,301 83,370
In Figures 5.10 to 5.12, the measured tie-lines with lysozyme can be compared with the ones
measured before, without lysozyme. For the linear and HBP, the tie-lines show no deviation from the
ones measured without lysozyme.
55
0 5 1 0 1 5 2 0 2 5 3 00
5
1 0
1 5
2 0
2 5
3 07 0
7 5
8 0
8 5
9 0
9 5
1 0 0
D e x t r a nW a t er
P E G
(a) 0,3 wt % of lysozyme
0 5 1 0 1 5 2 0 2 5 3 00
5
1 0
1 5
2 0
2 5
3 07 0
7 5
8 0
8 5
9 0
9 5
1 0 0
D e x t r a nW a t er
P E G
(b) 1 wt % of lysozyme
Figure 5.10: Experimental tie-lines of ATPS consisting of PEG 6000 – dextran 40 – water – lysozyme. Thevalues are presented in Tables B.16 to B.19.
0 5 1 0 1 5 2 0 2 5 3 00
5
1 0
1 5
2 0
2 5
3 07 0
7 5
8 0
8 5
9 0
9 5
1 0 0
D e x t r a nW a t er
P E G
(a) 0,3 wt % of lysozyme
0 5 1 0 1 5 2 0 2 5 3 00
5
1 0
1 5
2 0
2 5
3 07 0
7 5
8 0
8 5
9 0
9 5
1 0 0
D e x t r a nW a t er
P E G
(b) 1 wt % of lysozyme
Figure 5.11: Experimental tie-lines of ATPS consisting of PEG 8000 – dextran 40 – water – lysozyme. Thevalues are presented in Tables ?? to ??.
56
0 5 1 0 1 5 2 0 2 5 3 0 3 50
5
1 0
1 5
2 0
2 5
3 0
3 56 5
7 0
7 5
8 0
8 5
9 0
9 5
1 0 0
D e x t r a nW a t er
G 2
(a) ATPS consisting of G2
0 5 1 0 1 5 2 0 2 5 3 0 3 50
5
1 0
1 5
2 0
2 5
3 0
3 56 5
7 0
7 5
8 0
8 5
9 0
9 5
1 0 0
D e x t r a nW a t er
G 3
(b) ATPS consisting of G3
Figure 5.12: Experimental tie-lines of ATPS consisting of HBP – dextran 40 – water – lysozyme. The values arepresented in Tables ?? to ??.
The partition coefficient of the lysozyme, KP , was calculated with Equation (2.2) and the TLL
using Equation (2.1).
The obtained partition coefficients for ATPS with linear polymers as phase forming components
are presented in Figure 5.13, with the correspondent deviations. One possible way to interpret the
results is to consider that each pair of points illustrating the partitioning in systems containing the
same phase forming polymers and the same lysozyme concentration, has a tendency. This way, it is
possible to observe that for a lower protein concentration, 0,3 wt %, the partition coefficient increases
with the TLL, but for a higher protein concentration, 1 wt %, the opposite occurs. In both cases, more
partitioning is observed for an increasing TLL. Furthermore, the partitioning behaviour depends on
the molar weight of the PEG used, since in the systems formed by PEG 6000 the lysozyme partitions
to the bottom but in the case of PEG 8000 it partitions to the top.
57
1 7 1 8 1 9 2 0 2 1 2 2 2 3 2 4 2 5 2 6 2 7 2 80 , 8 00 , 8 50 , 9 00 , 9 51 , 0 01 , 0 51 , 1 01 , 1 51 , 2 01 , 2 5
Partit
ion
T L L
Figure 5.13: Partition coefficient experimentally obtained for ATPS formed by linear polymers PEG 6000 –Dextran (squares) and PEG 8000 – Dextran (triangles). The filled symbols correspond to systems with 0,3 wt %of lysozyme and the unfilled ones correspond to systems prepared with 1 wt % of lysozyme. The used data is
presented in Table B.28 of the Appendix.
It is known that the partition may be influenced by the concentrations and molecular mass of the
phase-forming polymer[65]. Moreover, the lysozyme solubility in PEG solutions decreases linearly
with the increase of PEG concentration, and the solubility is lower for a polyethylene glycol with a
higher degree of polymerization [66].
These studies are in agreement with the results obtained for the systems with 0,3 wt % of lysozyme
(filled symbols in Figure 5.13). For a higher concentration of PEG, longer TLL, a larger amount of
lysozyme partitions to the bottom phase, which can be explained by the solubility decrease in the
PEG phase. Moreover, the systems with PEG 8000 show a higher partitioning behaviour for a given
TLL: for a TLL of approximately 23 wt %, the PEG 8000 system has a KP of 1,15; for a similar TLL,
24,5 %, the lysozyme in the PEG 6000 system has a KP of 1,03.
However, the measured concentrations for the systems with 1 wt % of lysozyme were unexpected.
Although the protein shows a lower partitioning to the PEG phase when the polymer molar mass
is higher, the partition coefficient is lower than 1, which means that the lysozyme partitions to the
top phase (PEG) instead of the bottom. Also the KP decreases with longer TL, i.e., more lysozyme
partitions to the PEG phase where it has a higher amount of PEG. These results are not consistent
with the mentioned studies.
A reason for the unexpected results with a higher protein concentration may be the precipitation of
some of the protein in the interface of the systems. Since the solubility in the PEG solution was verified
in literature, a possible explanation may be a low solubility in the dextran phase. This would cause
58
the protein to precipitate when in contact with that phase, lowering the partitioning coefficient. This
effect would be more significant as the TLL increases, since then the dextran concentration increases
as well. Furthermore, in Figure 5.13 it is possible to observe a tendency of the points with higher
protein concentration, which suggests an independence from the PEG molar mass, and agrees with
the given interpretation.
Knowing the partitioning behaviour of lysozyme, a modelling attempt was made. A quaternary
LCT model in combination with the Wertheim theory was used. At this stage, the studied ATPS are
already well described by the model with the adjusted parameters presented in Section 5.2, and the
interaction and association parameters of lysozyme in a water solution are adjusted as well, and can
be found in Section 5.1. Therefore, the interaction between the lysozyme and the phase forming
polymers is what still needs to be modelled.
For each system, PEG 6000 – dextran – water and PEG 8000 – dextran – water, the partition for
one of the tie lines was adjusted. To test the model, the partitioning for the second tie line was calcu-
lated and compared to the measured value. In Figure 5.14 the results for a lysozyme concentration
of 0,3 wt % are presented.
1 5 2 0 2 5 3 0
1 , 0
1 , 1
1 , 2
Partit
ion co
efficie
nt
T L L
Figure 5.14: Partition coefficient experimentally obtained for ATPS formed by linear polymers PEG 6000 –Dextran – water and PEG 6000 – Dextran – water (filled symbols), and values calculated by the LCT+Wertheim
(unfilled symbols), for a lysozyme concentration of 0,3 wt %.
Comparing the modelling results with the experimental data regarding the systems with 0,3 wt %
of lysozyme, the model can be considered to describe the partitioning behaviour of the lysozyme in
good accordance with the experimental data. Therefore, using the LCT+Wertheim model, the phase
behaviour of quaternary systems can be described with an acceptable accuracy.
In Figure 5.15, the results for a lysozyme concentration of 1 wt % are presented. The calculated
59
results do not fit the ones obtained experimentally, which can be explained by the already mentioned
protein’s precipitation. In this work, the lysozyme concentration in the systems was chosen based
on literature information in comparable systems. In order to test the model used for higher lysozyme
concentrations, solubility measurements should be made for the conditions used in this work.
1 5 2 0 2 5 3 00 , 8
0 , 9
1 , 0
1 , 1
1 , 2
Partit
ion co
efficie
nt
T L L
Figure 5.15: Partition coefficient experimentally obtained for ATPS formed by linear polymers PEG 6000 –Dextran – water and PEG 6000 – Dextran – water (filled symbols), and values calculated by the LCT+Wertheim
(unfilled symbols), for a lysozyme concentration of 1 wt %.
Regarding the ATPS formed by HBP, a lot of precipitation was observed, and for that reason the
obtained results were not modelled, since they do not represent the real behaviour of lysozyme in
the ATPS based on G2 and G3. This problem could be solved by a test on the lysozyme solubility
in solutions with G2 and G3 polymers. The calculated lysozyme concentrations are presented in
Table 5.8
Table 5.8: Absorbance and lysozyme concentration on the top and bottom phases of ATPS containing G2. TheA correspond to systems with 0,3 wt % of lysozyme and the B to systems with 1 wt %.
A B
System 1 System 2 System 1
Top Bottom Top Bottom Top Bottom
Ameas 0,655 0,344 0,592 0,390 0,512 0,206G2 wt% 4,77E-02 6,60E-03 3,71E-02 1,32E-02 2,46E-02 3,24E-03Acalc 0,569 0,079 0,442 0,157 0,293 0,039Aest 0,086 0,265 0,150 0,233 0,219 0,167
Lysozyme wt % 0,110 0,247 0,246 0,218 1,02 0,304
As expected, in five out of the six separated phases the concentrations are lower than in the initial
prepared mixtures (Table 3.17), result of the protein precipitation. Given the obtained results for these
60
systems, no further measurements were made for ATPS containing HBP.
61
62
6Conclusions and Future Work
63
Within this work, the phase behaviour of ATPS composed of linear polymer (PEG 6000/PEG
8000/dextran) was analysed, as well as the novel ATPS containing branched polymers (G2/G3). The
LLE diagrams of the systems were determined by the investigation of the binodal curves and tie-lines.
The partitioning behaviour of lysozyme in the PEG 6000 – dextran – water system and PEG 8000
– dextran – water systems was investigated. Additionally, the quaternary systems containing both
lysozyme and HBP were also investigated.
With all the experimental results of the LLE, a thermodynamic calculation model with Lattice Clus-
ter Theory (LCT) combined with Wertheim theory was adjusted to describe the equilibrium of the
systems with linear and branched polymers, as well as the partitioning of the lysozyme in the linear
systems.
Parameters in the model were determined in order for the calculated results to match the experi-
mental data in the most precise way.
The obtained tie-lines have acceptable deviations to the demixing points. Reasons for these neg-
ligible deviations can be the non-complete phase separation and the non-optimized HPLC method.
Comparing the experimental and modelled results regarding the binodal curves for systems contain-
ing linear and branched polymers, it can be seen that the tie-lines of the systems containing branched
polymers have steeper tie lines, which means that the minimum required amount of polymers to form
the system is higher when using HBP. This is one disadvantage of using HBP as phase forming
components. Additionally G2 and G3 polymers are more expensive than the used linear polymers.
Therefore, in case the HBP based systems show an improvement on the partitioning of a product, an
economic analysis would have to be made in order to understand if the improvement is worth the extra
cost. Moreover, the viscosity of the formed phases has an important influence in the phase separation
time, thus viscosity measurements should be made in order to better optimize the process.
Modelling with LCT and Wertheim theory for binodal curves and tie lines can achieve a good
accordance between experimental and calculated data for the systems containing linear polymers. In
this case the association parameters are constant for both PEG 6000 and PEG 8000, and based on
literature, and the structural parameters were calculated taking into account the molar mass of the
polymers. The interaction parameters were adjusted for each polymer and compared with the ones
calculated through equations that describe a relation between the interaction parameters and the
molar mass of the interacting polymers, defined by Kulaguin Chicaroux and Zeiner. The comparison
showed that the most important deviation corresponded to the dextran – water interaction, which can
be a result of the polydispersity of this polymer.
The modelling of the systems containing HBP was made by adjusting all the interaction and as-
sociative parameters related with the G2 and G3 polymers to the obtained experimental data. The
structural parameters were defined by the polymers’ structure and the segment size was determined
based on the system containing G3, in order to adjust the tie-line slope to the experimental results.
This parameter was also used to calculate the G2 based ATPS. The binodal curve was well fitted in
both systems, although the tie-lines for the G2 system showed some deviation to the experimentally
obtained data. This can be explained by the polydispersity of the HBP, which is not considered in the
64
model.
The lysozyme partitioning in systems containing linear polymers with a low protein concentration
was also successfully modelled, using a quaternary LCT+Wertheim model. The interaction and as-
sociation parameters between lysozyme and water were adjusted to literature data. The structural
parameters were defined considering the biomolecule as a linear polymer with an amino acid per
segment. The interaction and association parameters between the lysozyme and the polymers were
determined by adjustment of the partitioning to the experimental data. The modelled partitioning
behaviour was in good accordance with the experimental data.
It was not possible to measure the lysozyme concentration in the HBP based systems, because
a part of the protein precipitated. In order to better study and compare the partitioning behaviour of
biomolecules in ATPS containing HBP, solubility measurements should be made. The binodal curve
and tie-lines were not influenced by the presence of lysozyme in any of the systems.
In further research, dispersity of the polymer can be considered in the model. This way, a higher
accuracy could be achieved. Furthermore, as in the past [13], the model was successfully applied to
calculate linear polymers based systems. Moreover in the present work it was also applied to HBP
based systems. Therefore, it can be considered that the model has great potential to help with a more
economically favourable development of novel ATPS.
65
66
Bibliography
[1] F. M. Wurm, “Production of recombinant protein therapeutics in cultivated mammalian cells,”
Nature biotechnology, vol. 22, no. 11, pp. 1393–1398, 2004.
[2] R. A. Rader, “Fda biopharmaceutical product approvals and trends in 2012: Up from 2011,
but innovation and impact are limited,” BioProcess International, vol. 11, no. 3, pp. 18–27,
2013. [Online]. Available: http://www.bioprocessintl.com/manufacturing/antibody-non-antibody/
fda-biopharmaceutical-product-approvals-and-trends-in-2012-340619/
[3] A. Straathof, “The proportion of downstream costs in fermentative production processes,” Com-
prehensive biotechnology, vol. 2, pp. 811–814, 2011.
[4] U. Gottschalk, “Bioseparation in antibody manufacturing: the good, the bad and the ugly,”
Biotechnology progress, vol. 24, no. 3, pp. 496–503, 2008.
[5] E. M. Agency, “Guideline on setting specifications for related impurities in antibiotics,” 2012.
[6] T. Reschke, Modeling and design of aqueous two-phase systems, 1st ed., ser. Schriftenreihe
Thermodynamik. Munchen: Verl. Dr. Hut, 2014, vol. 11.
[7] B. Y. Zaslavsky, Aqueous two-phase partitioning: Physical chemistry and bioanalytical applica-
tions, 1st ed. New York: Dekker, 1995.
[8] Rosa, P A J, I. F. Ferreira, A. M. Azevedo, and M. R. Aires-Barros, “Aqueous two-phase sys-
tems: A viable platform in the manufacturing of biopharmaceuticals,” Journal of chromatogra-
phy. A, vol. 1217, no. 16, pp. 2296–2305, 2010.
[9] A. Prinz, K. Koch, A. Gorak, and T. Zeiner, “Multi-stage laccase extraction and separation using
aqueous two-phase systems: Experiment and model,” Process Biochemistry, vol. 49, no. 6, pp.
1020–1031, 2014.
[10] Costa, Maria Joao L., M. T. Cunha, J. M. Cabral, and M. R. Aires-Barros, “Scale-up of recom-
binant cutinase recovery by whole broth extraction with peg-phosphate aqueous two-phase,”
Bioseparation, vol. 9, no. 4, pp. 231–238, 2000.
[11] J. A. Asenjo and B. A. Andrews, “Aqueous two-phase systems for protein separation: A per-
spective,” Journal of Chromatography A, vol. 1218, no. 49, pp. 8826–8835, 2011.
67
[12] J. Persson, H.-O. Johansson, I. Galaev, B. Mattiasson, and F. Tjerneld, “Aqueous polymer two-
phase systems formed by new thermoseparating polymers,” Bioseparation, vol. 9, no. 2, pp.
105–116, 2000.
[13] A. Kulaguin-Chicaroux and T. Zeiner, “Novel aqueous two-phase system based on a hyper-
branched polymer,” Fluid Phase Equilibria, vol. 362, pp. 1–10, 2014.
[14] M. van Berlo, K. C. Luyben, and van der Wielen, Luuk A.M, “Poly(ethylene glycol)–salt aqueous
two-phase systems with easily recyclable volatile salts,” Journal of Chromatography B: Biomed-
ical Sciences and Applications, vol. 711, no. 1-2, pp. 61–68, 1998.
[15] A. Kumar, A. Srivastava, I. Y. Galaev, and B. Mattiasson, “Smart polymers: Physical forms
and bioengineering applications,” Progress in Polymer Science, vol. 32, no. 10, pp. 1205–1237,
2007.
[16] M. R. Aguilar and J. Roman, Smart polymers and their applications. Cambridge: Woodhead
Publishing, 2014.
[17] D. Othmer and P. Tobias, “Liquid-liquid extraction data- partial pressures of ternary liquid sys-
tems and the prediction of tie lines,” Industrial & Engineering Chemistry, vol. 34, no. 6, pp.
696–700, 1942.
[18] P. Gonzalez-Tello, F. Camacho, G. Blazquez, and F. J. Alarcon, “Liquid−liquid equilibrium in the
system poly(ethylene glycol) + mgso 4 + h 2 o at 298 k,” Journal of Chemical & Engineering
Data, vol. 41, no. 6, pp. 1333–1336, 1996.
[19] Edmond E, Ogston AG., “An approach to the study of phase separation in ternary aqueous
systems,” Biochemical Journal, no. 109(4), pp. 569–576, 1968.
[20] Edmond E, Ogston AG, “Phase separation in an aqueous quaternary system,” Biochemical
Journal, no. 117, pp. 85–89, 1970.
[21] E. A. Muller and K. E. Gubbins, “Molecular-based equations of state for associating fluids: A
review of saft and related approaches,” Industrial & Engineering Chemistry Research, vol. 40,
no. 10, pp. 2193–2211, 2001.
[22] A. K. Chicaroux and T. Zeiner, “Investigation of interfacial properties of aqueous two-phase
systems by density gradient theory,” Fluid Phase Equilibria, 2015.
[23] P. J. Flory, Principles of polymer chemistry, 12th ed. Ithaca: Cornell Univ. Pr, 1983.
[24] M. L. Huggins, “Theory of solutions of high polymers 1,” Journal of the American Chemical
Society, vol. 64, no. 7, pp. 1712–1719, 1942.
[25] J. Dudowicz, K. F. Freed, and W. G. Madden, “Role of molecular structure on the thermody-
namic properties of melts, blends, and concentrated polymer solutions: comparison of monte
68
carlo simulations with the cluster theory for the lattice model,” Macromolecules, vol. 23, no. 22,
pp. 4803–4819, 1990.
[26] J. Dudowicz and K. F. Freed, “Effect of monomer structure and compressibility on the proper-
ties of multicomponent polymer blends and solutions: 1. lattice cluster theory of compressible
systems,” Macromolecules, vol. 24, no. 18, pp. 5076–5095, 1991.
[27] J. Dudowicz, K. F. Freed, and J. F. Douglas, “Modification of the phase stability of polymer
blends by diblock copolymer additives,” Macromolecules, vol. 28, no. 7, pp. 2276–2287, 1995.
[28] K. F. Freed, Ed., Phase Behaviour of Polymer Blends, ser. Advances in Polymer Science.
Berlin/Heidelberg: Springer-Verlag, 2005.
[29] K. F. Freed and J. Dudowicz, “Influence of monomer molecular structure on the miscibility of
polymer blends,” in Phase Behaviour of Polymer Blends, ser. Advances in Polymer Science,
K. F. Freed, Ed. Berlin/Heidelberg: Springer-Verlag, 2005, vol. 183, pp. 63–126.
[30] M. S. Wertheim, “Fluids with highly directional attractive forces. i. statistical thermodynamics,”
Journal of Statistical Physics, no. 1-2, pp. 19–34, 1984. [Online]. Available: http:
//link.springer.com/journal/10955
[31] M. S. Wertheim, “Fluids with highly directional attractive forces. ii. thermodynamic perturbation
theory and integral equations,” Journal of Statistical Physics, vol. 35, no. 1-2, pp. 35–47, 1984.
[32] L.-H. Mei, D.-Q. Lin, Z.-Q. Zhu, and Z.-X. Han, “Densities and viscosities of polyethylene glycol
+ salt + water systems at 20 c,” J. Chem. Eng. Data, vol. 40, no. 6, pp. 1168–1171, 1995.
[33] M. Beijerinck, “Ueber emulsionsbildung bei der vermischung wasseriger losungen gewisser
gelatinierender kolloide,” Colloid & Polymer Science, vol. 7, no. 1, pp. 16–20, 1910.
[34] P. A. Albertsson, “Separation of cell particles and molecules: A citation-classic commentary
on partition of cell particles and macromolecules by albertsson, p.a,” Current Contents / Life
Sciences, vol. 22, 1990.
[35] P. Ake Albertsson, “Partition of proteins in liquid polymer-polymer two-phase systems,” Nature,
vol. 182, pp. 709–711, 1958. [Online]. Available: http://www.nature.com/nature/journal/v182/
n4637/abs/182709a0.html
[36] P. Ake Albertsson, “Particle fractionation in liquid two-phase systems the composition of some
phase systems and the behaviour of some model particles in them application to the isolation
of cell walls from microorganisms,” Biochimica et Biophysica Acta, vol. 27, pp. 378 – 395,
1958. [Online]. Available: http://www.sciencedirect.com/science/article/pii/0006300258903457
[37] T. J. Peters, “Partition of cell particles and macromolecules: Separation and purification of
biomolecules, cell organelles, membranes and cells in aqueous polymer two phase systems
and their use in biochemical analysis and biotechnology. p-a. albertsson. third edition, 1986,
69
john wiley and sons, chichester, A£61.35 pages 346.” Cell Biochemistry and Function, vol. 5,
no. 3, pp. 233–234, 1987. [Online]. Available: http://dx.doi.org/10.1002/cbf.290050311
[38] K. H. Kroner, H. Hustedt, S. Granda, and M.-R. Kula, “Technical aspects of separation
using aqueous two-phase systems in enzyme isolation processes,” Biotechnology and
Bioengineering, vol. 20, no. 12, pp. 1967–1988, 1978. [Online]. Available: http:
//dx.doi.org/10.1002/bit.260201211
[39] H. Water, D. E. Brooks, and D. Fisher, Partitioning in aqueous two-phase systems: Theory,
methods, uses, and applications to biotechnology. Orlando: Academic Press, 1985.
[40] P. A. J. Rosa, A. M. Azevedo, S. Sommerfeld, M. Mutter, W. Backer, and M. R. Aires-Barros,
“Continuous purification of antibodies from cell culture supernatant with aqueous two-phase
systems: From concept to process,” Biotechnology Journal, vol. 8, no. 3, pp. 352–362, 2013.
[Online]. Available: http://dx.doi.org/10.1002/biot.201200031
[41] M. Rito-Palomares, “Practical application of aqueous two-phase partition to process
development for the recovery of biological products,” Journal of Chromatography B, vol. 807,
no. 1, pp. 3 – 11, 2004, 12th International Conference on Biopartitioning and Purification.
[Online]. Available: http://www.sciencedirect.com/science/article/pii/S1570023204000467
[42] B. Andrews and J. Asenjo, “Protein partitioning equilibrium between the aqueous poly(ethylene
glycol) and salt phases and the solid protein phase in poly(ethylene glycol)-salt two-phase
systems,” Journal of chromatography. B, Biomedical applications, vol. 685, no. 1, pp. 15–20,
October 1996. [Online]. Available: http://dx.doi.org/10.1016/0378-4347(96)00134-X
[43] R. H. Kienle and A. Hovey, “The polyhydric alcohol-polybasic acid reaction. i. glycerol-phthalic
anhydride,” Journal of the American Chemical Society, vol. 51, no. 2, pp. 509–519, 1929.
[44] G. Odian, Principles of polymerization. John Wiley & Sons, 2004.
[45] P. J. Flory, “Molecular size distribution in three dimensional polymers. i. gelation1,” Journal of
the American Chemical Society, vol. 63, no. 11, pp. 3083–3090, 1941.
[46] v. J. Flory, “Molecular size distribution in three dimensional polymers. ii. trifunctional branching
units,” Journal of the American Chemical Society, vol. 63, no. 11, pp. 3091–3096, 1941.
[47] v. P. J. Flory, “Molecular size distribution in three dimensional polymers. iii. tetrafunctional
branching units,” Journal of the American Chemical Society, vol. 63, no. 11, pp. 3096–3100,
1941.
[48] Y. H. Kim and O. W. Webster, “Water soluble hyperbranched polyphenylene:” a unimolecular
micelle?”,” Journal of the American Chemical Society, vol. 112, no. 11, pp. 4592–4593, 1990.
[49] C. Gao and D. Yan, “Hyperbranched polymers: from synthesis to applications,” Progress in
Polymer Science, vol. 29, no. 3, pp. 183–275, 2004.
70
[50] D. A. Tomalia and J. M. Frechet, “Discovery of dendrimers and dendritic polymers: A brief
historical perspective*,” Journal of Polymer Science Part A: Polymer Chemistry, vol. 40, no. 16,
pp. 2719–2728, 2002.
[51] H. Stutz, “The glass temperature of dendritic polymers,” Journal of Polymer Science Part B:
Polymer Physics, vol. 33, no. 3, pp. 333–340, 1995.
[52] A. V. Lyulin, D. B. Adolf, and G. R. Davies, “Computer simulations of hyperbranched polymers
in shear flows,” Macromolecules, vol. 34, no. 11, pp. 3783–3789, 2001.
[53] T. H. Mourey, S. Turner, M. Rubinstein, J. Frechet, C. Hawker, and K. Wooley, “Unique behav-
ior of dendritic macromolecules: intrinsic viscosity of polyether dendrimers,” Macromolecules,
vol. 25, no. 9, pp. 2401–2406, 1992.
[54] E. Malmstrom and A. Hult, “Hyperbranched polymers,” Journal of Macromolecular Science, Part
C: Polymer Reviews, vol. 37, no. 3, pp. 555–579, 1997.
[55] K. Matyjaszewski, T. Shigemoto, J. M. Frechet, and M. Leduc, “Controlled/living radical poly-
merization with dendrimers containing stable radicals,” Macromolecules, vol. 29, no. 12, pp.
4167–4171, 1996.
[56] S.-E. Stiriba, H. Frey, and R. Haag, “Dendritic polymers in biomedical applications: from poten-
tial to clinical use in diagnostics and therapy,” Angewandte Chemie International Edition, vol. 41,
no. 8, pp. 1329–1334, 2002.
[57] J. L. Hedrick, C. J. Hawker, R. D. Miller, R. Twieg, S. Srinivasan, and M. Trollsas, “Structure
control in organic-inorganic hybrids using hyperbranched high-temperature polymers,” Macro-
molecules, vol. 30, no. 24, pp. 7607–7610, 1997.
[58] Y. H. Kim and O. W. Webster, “Hyperbranched polyphenylenes,” Macromolecules, vol. 25,
no. 21, pp. 5561–5572, 1992.
[59] M. K. Kozlowska, B. F. Jurgens, C. S. Schacht, J. Gross, and T. W. de Loos, “Phase behavior of
hyperbranched polymer systems: experiments and application of the perturbed-chain polar saft
equation of state,” The Journal of Physical Chemistry B, vol. 113, no. 4, pp. 1022–1029, 2009.
[60] J. G. Jang and Y. C. Bae, “Phase behavior of hyperbranched polymer solutions with specific
interactions,” The Journal of Chemical Physics, vol. 114, no. 11, pp. 5034–5042, 2001.
[61] L. Lue and J. M. Prausnitz, “Structure and thermodynamics of homogeneous-dendritic-polymer
solutions: computer simulation, integral-equation, and lattice-cluster theory,” Macromolecules,
vol. 30, no. 21, pp. 6650–6657, 1997.
[62] M. Seiler, J. Rolker, L. Mokrushina, H. Kautz, H. Frey, and W. Arlt, “Vapor–liquid equilibria in
dendrimer and hyperbranched polymer solutions: experimental data and modeling using unifac-
fv,” Fluid phase equilibria, vol. 221, no. 1, pp. 83–96, 2004.
71
[63] S. D. Flanagan and S. H. Barondes, “Affinity partitioning. a method for purification of proteins
using specific polymer-ligands in aqueous polymer two-phase systems.” Journal of Biological
Chemistry, vol. 250, no. 4, pp. 1484–1489, 1975.
[64] T. Franco, A. Andrews, and J. Asenjo, “Conservative chemical modification of proteins to study
the effects of a single protein property on partitioning in aqueous two-phase systems,” Biotech-
nology and bioengineering, vol. 49, no. 3, pp. 290–299, 1996.
[65] U. Gunduz, “Partitioning of bovine serum albumin in an aqueous two-phase system: optimiza-
tion of partition coefficient,” Journal of Chromatography B: Biomedical Sciences and Applica-
tions, vol. 743, no. 1, pp. 259–262, 2000.
[66] M. Boncina, J. Rescic, and V. Vlachy, “Solubility of lysozyme in polyethylene glycol-electrolyte
mixtures: the depletion interaction and ion-specific effects,” Biophysical journal, vol. 95, no. 3,
pp. 1285–1294, 2008.
[67] G. Tubio, B. Nerli, and G. Pico, “Relationship between the protein surface hydrophobicity and its
partitioning behaviour in aqueous two-phase systems of polyethyleneglycol–dextran,” Journal
of Chromatography B, vol. 799, no. 2, pp. 293–301, 2004.
[68] J. W. Gibbs, “On the equilibrium of heterogeneous substances,” American Journal of Science,
vol. s3-16, no. 96, pp. 441–458, 1878.
[69] D. Othmer and P. Tobias, “Liquid-liquid extraction data - the line correlation,” Industrial
& Engineering Chemistry, vol. 34, no. 6, pp. 693–696, 1942. [Online]. Available:
http://dx.doi.org/10.1021/ie50390a600
[70] P. Gonzalez-Tello, F. Camacho, G. Blazquez, , and F. J. Alarcon, “Liquid-liquid equilibrium in the
system poly(ethylene glycol) + mgso4 + h2o at 298 k,” Journal of Chemical & Engineering Data,
vol. 41, no. 6, pp. 1333–1336, 1996. [Online]. Available: http://dx.doi.org/10.1021/je960075b
[71] M. Hu, Q. Zhai, Y. Jiang, L. Jin, , and Z. Liu, “Liquid-liquid and liquid-liquid-solid equilibrium in
peg + cs2so4 + h2o,” Journal of Chemical & Engineering Data, vol. 49, no. 5, pp. 1440–1443,
2004. [Online]. Available: http://dx.doi.org/10.1021/je0498558
[72] M. Jayapal, I. Regupathi, , and T. Murugesan, “Liquid-liquid equilibrium of poly(ethylene glycol)
2000 + potassium citrate + water at (25, 35, and 45) c,” Journal of Chemical & Engineering
Data, vol. 52, no. 1, pp. 56–59, 2007. [Online]. Available: http://dx.doi.org/10.1021/je060209d
[73] W. G. McMillan Jr and J. E. Mayer, “The statistical thermodynamics of multicomponent systems,”
The Journal of Chemical Physics, vol. 13, no. 7, pp. 276–305, 1945.
[74] M. Valavi, M. R. Dehghani, and F. Feyzi, “Calculation of liquid-liquid equilibrium in polymer
electrolyte solutions using phsc-electrolyte equation of state,” Fluid Phase Equilibria, vol.
341, pp. 96 – 104, 2013. [Online]. Available: http://www.sciencedirect.com/science/article/pii/
S0378381212005870
72
[75] T. Reschke, C. Brandenbusch, and G. Sadowski, “Modeling aqueous two-phase systems:
I. polyethylene glycol and inorganic salts as ATPS former,” Fluid Phase Equilibria, vol.
368, pp. 91 – 103, 2014. [Online]. Available: http://www.sciencedirect.com/science/article/pii/
S0378381214000995
[76] M. Margules, “Uber die zusammensetzung’ der gesattigten dampfe von mischungen,”
Sitzungsberichte der Kaiserliche Akadamie der Wissenschaften Wien Mathematisch-
Naturwissenschaftliche Klasse II, vol. 104, pp. 1243–1278, 1895. [Online]. Available:
https://archive.org/details/sitzungsbericht10wiengoog
[77] C. H. Kang and S. I. Sandler, “A thermodynamic model for two-phase aqueous polymer
systems,” Biotechnology and Bioengineering, vol. 32, no. 9, pp. 1158–1164, 1988. [Online].
Available: http://dx.doi.org/10.1002/bit.260320909
[78] C. Grossmann, J. Zhu, and G. Maurer, “Phase equilibrium studies on aqueous two-phase
systems containing amino acids and peptides,” Fluid Phase Equilibria, vol. 82, pp. 275 – 282,
1993. [Online]. Available: http://www.sciencedirect.com/science/article/pii/037838129387151P
[79] M. T. Zafarani-Moattar and R. Sadeghi, “Measurement and correlation of liquid-liquid equilibria
of the aqueous two-phase system polyvinylpyrrolidone-sodium dihydrogen phosphate,”
Fluid Phase Equilibria, vol. 203, no. 1-2, pp. 177 – 191, 2002. [Online]. Available:
http://www.sciencedirect.com/science/article/pii/S0378381202001796
[80] C.-C. Chen and L. B. Evans, “A local composition model for the excess gibbs energy of
aqueous electrolyte systems,” AIChE Journal, vol. 32, no. 3, pp. 444–454, 1986. [Online].
Available: http://dx.doi.org/10.1002/aic.690320311
[81] G. M. Kontogeorgis, A. Fredenslund, and D. P. Tassios, “Simple activity coefficient model for
the prediction of solvent activities in polymer solutions,” Industrial & engineering chemistry re-
search, vol. 32, no. 2, pp. 362–372, 1993.
[82] R. H. Petrucci, W. S. Harwood, G. E. Herring, and J. Madura, General Chemistry: Principles
and Modern Application, 9th ed. Prentice Hall, Inc, 2006.
[83] P. J. Flory, “Thermodynamics of high polymer solutions,” The Journal of Chemical Physics,
vol. 10, no. 1, p. 51, 1942.
[84] M. L. Huggins, “Solutions of long chain compounds,” The Journal of Chemical Physics, vol. 9,
no. 5, p. 440, 1941.
[85] H. Cabezas, “Theory of phase formation in aqueous two-phase systems,” Journal of Chro-
matography B: Biomedical Sciences and Applications, vol. 680, no. 1-2, pp. 3–30, 1996.
[86] R. Koningsveld and L. Kleintjens, “Liquid-liquid phase separation in multicomponent poly-
mer systems. x. concentration dependence of the pair-interaction parameter in the system
cyclohexane-polystyrene,” Macromolecules, vol. 4, no. 5, pp. 637–641, 1971.
73
[87] D. Patterson, “Free volume and polymer solubility. a qualitative view,” Macromolecules, vol. 2,
no. 6, pp. 672–677, 1969.
[88] (2015, July) Online catalog of hyperbranched peg products offered by polymer factory.
[Online]. Available: http://www.polymerfactory.com/multifunctional-pegs/hyperbranched-pegs/
hyperbranched-peg-6kda
[89] R. Hatti-Kaul, Aqueous Two-Phase Systems: Methods and Protocols (Methods in Biotechnol-
ogy), 2000th ed. Humana Press, 2000.
[90] S. Mori and H. G. Barth, Size Exclusion Chromatography. Springer, 1999.
[91] P. Hong, S. Koza, and E. S. Bouvier, “A review size-exclusion chromatography for the analysis
of protein biotherapeutics and their aggregates,” Journal of liquid chromatography & related
technologies, vol. 35, no. 20, pp. 2923–2950, 2012.
[92] K. Im, H. woong Park, S. Lee, and T. Chang, “Two-dimensional liquid chromatography analysis
of synthetic polymers using fast size exclusion chromatography at high column temperature,”
Journal of Chromatography A, vol. 1216, no. 21, pp. 4606 – 4610, 2009. [Online]. Available:
http://www.sciencedirect.com/science/article/pii/S0021967309004828
[93] M. Boncina, J. Rescic, and V. Vlachy, “Solubility of lysozyme in polyethylene glycol-electrolyte
mixtures: The depletion interaction and ion-specific effects,” Biophysical Journal, vol. 95, no. 3,
pp. 1285–1294, 2008.
[94] S. B. Howard, P. J. Twigg, J. K. Baird, and E. J. Meehan, “The solubility of hen egg-white
lysozyme,” Journal of Crystal Growth, vol. 90, no. 1, pp. 94–104, 1988.
[95] D. E. Kuehner, J. Engmann, F. Fergg, M. Wernick, H. W. Blanch, and J. M. Prausnitz, “Lysozyme
net charge and ion binding in concentrated aqueous electrolyte solutions,” The Journal of Phys-
ical Chemistry B, vol. 103, no. 8, pp. 1368–1374, 1999.
[96] P. Atkins and J. De Paula, Physical chemistry for the life sciences. Oxford University Press,
2011.
[97] V. Lee, Peptide and protein drug delivery. CRC Press, 1990, vol. 4.
[98] S. Khosharay, M. Abolala, and F. Varaminian, “Modeling the surface tension and surface prop-
erties of (co 2+ h 2 o) and (h 2 s+ h 2 o) with gradient theory in combination with spc–saft eos
and a new proposed influence parameter,” Journal of Molecular Liquids, vol. 198, pp. 292–298,
2014.
[99] K. Langenbach and S. Enders, “Development of an eos based on lattice cluster theory for pure
components,” Fluid Phase Equilibria, vol. 331, pp. 58–79, 2012.
[100] I. G. Economou and M. D. Donohue, “Chemical, quasi-chemical and perturbation theories for
associating fluids,” AIChE Journal, vol. 37, no. 12, pp. 1875–1894, 1991.
74
[101] S. Enders, K. Langenbach, P. Schrader, and T. Zeiner, “Phase diagrams for systems containing
hyperbranched polymers,” Polymers, vol. 4, no. 1, pp. 72–115, 2012. [Online]. Available:
http://www.mdpi.com/2073-4360/4/1/72/htm
[102] K. Langenbach and S. Enders, “Development of an EOS based on lattice cluster theory
for pure components,” Fluid Phase Equilibria, vol. 331, pp. 58 – 79, 2012. [Online]. Available:
http://www.sciencedirect.com/science/article/pii/S0378381212002774
[103] P. Databank, “hen egg-white lysozyme at 2.0 a resolution,” 2013, online accessed September,
2015. [Online]. Available: http://www.rcsb.org/pdb/images/2cds asr r 500.jpg
[104] V. G. Taratuta, A. Holschbach, G. M. Thurston, D. Blankschtein, and G. B. Benedek, “Liquid-
liquid phase separation of aqueous lysozyme solutions: effects of ph and salt identity,” The
Journal of Physical Chemistry, vol. 94, no. 5, pp. 2140–2144, 1990.
[105] A. K. Chicaroux and T. Zeiner, “Investigation of interfacial properties of aqueous two-phase
systems by density gradient theory,” Fluid Phase Equilibria, 2015.
75
76
ALCT correction terms
XS1,kλ =
(1− 1
Mk
) [1Mk− 1
Mλ
](A.1)
XS2,kλ =
(1− 1
Mk
) [1Mk
(2b3,k + 6b4,k − 3)− 1Mλ
(2b3,λ + 6b4,λ − 3)]
(A.2)
XS3,kλ =
(1− 2
Mk+
b3,kMk
+3b4,kMk
) [1Mk
(2− b3,k − 3b4,k)− 1Mλ
(2− b3,λ − 3b4,λ)]
(A.3)
XS4,kλµ =
(M2k−5Mk−2b3,k−6b4,k+6
2Mk
) [1
MλMµ− 1
Mλ− 1
Mµ− 1
M2k
+ 2Mk
](A.4)
XS5,kλµη =
(1− 1
Mk
) [(3−3Mµ−10Mη)Mλ+(3−10Mη)Mµ+10Mη−3
3MλMµMη+
(Mλ+Mµ−1)Mk
MλMµ+(
263 −
163Mk
+ 1M2k
)1Mk− 2
(A.5)
Xε1,kλµ = 1
Mλ
(1− 1
Mk
) [2Mλ−2Mµ
+ 2b3,λ + 6b4,λ −Mλ − 1]
(A.6)
Xε2,kλ = 2
Mλ
(1− 1
Mk
)[2Mλ − b3,λ − 3b4,λ −MkMλ +Mk − 1] +
2b3,k+6b4,k−3Mk
+ 1 (A.7)
Xε3,kλ = 1
Mλ
(1− 1
Mk
)[2b3,λ + 6b4,λ −Mk + (Mk −Mλ + 2)Mλ − 3] +
3−2b3,k−6b4,k2Mk
− 12 (A.8)
Xε4,kλ = 2
Mλ
(1− 1
Mk
) [4− b3,λ − 3b4,λ +M2
λ − 4Mλ
](A.9)
Xε5,kλµ = 1
Mλ
(1− 1
Mk
) [1−Mλ−Mµ
Mµ+Mλ
](A.10)
Xε6,k =
(1− 1
Mk
)(A.11)
Xε2
1,k = 1Mk
(b3,k + 3b4,k − 1) (A.12)
Xε2
2,kλ = 1Mk
(2Mλ− b3,k − 3b4,k − 2Mk
Mλ+ 2Mk − 1
)(A.13)
Xε2
3,k = 1Mk
(2Mk − b3,k − 3b4,k − 2Mk
Mλ+ 2
Mλ− 1)
(A.14)
Xε2
4,kλ = 1Mk
(3Mλ− b3,k − 3b4,k − 3Mk
Mλ+ 7Mk
2 − 52
)(A.15)
A-1
Xε2
5,kλ = 1Mk
(b3,k + 3b4,k + 2Mk
Mλ− 2
Mλ− 3Mk + 2
)(A.16)
Xε2
6,kλ = 1Mk
(b3,k + 3b4,k − 4Mk + 4Mk
Mλ− 4
Mλ+ 3)
(A.17)
Xε2
7,kλ = 1Mk
(Mk − Mk+Mλ−1
Mλ
)(A.18)
Xε2
8,k = 1Mk
(Mk − 1) (A.19)
A-2
BExperimental data
B.0.3 Calibration
Table B.1: Calibration data for the linear polymers.
PEG6000 PEG8000 Dextran
Concentration(wt%) Area Concentration
(wt%) Area Concentration(wt%) Area
0,20555 5,60165 0,20124 5,820550,20555 5,61567 0,20286 5,56172 0,20124 5,887360,10299 2,57547 0,09907 2,46163 0,10059 2,943670,10299 2,55205 0,09907 2,48987 0,10059 2,942880,05166 1,06199 0,04979 1,01198 0,04991 1,471150,05166 1,10589 0,04979 1,00754 0,04991 1,524040,02589 0,38414 0,02459 0,29426 0,00988 0,325860,02589 0,35823 0,02459 0,29983 0,00988 0,292670,15034 4,09049 0,15177 3,87341 0,29827 8,72340,15034 4,11892 0,15177 4,02578 0,29827 8,694760,01004 0 0,01012 0,114890,01004 0,09347 0,01012 0
B-1
Table B.2: Calibration data for the HBP.
G2 G3
Concentration(wt%) Area Concentration
(wt%) Area
0,20478 3,27717 0,21071 3,782680,20478 3,55831 0,21071 4,028860,1026 1,67679 0,10004 1,820570,1026 1,67147 0,10004 1,80323
0,05092 0,74955 0,05261 0,900820,05092 0,75902 0,05261 0,893120,01106 0,16147 0,01082 0,173930,01106 0,14277 0,01082 0,17628
Table B.3: Calibration data for lysozyme.
Concentration(%wt) Absorbance
0,01073 1,56260,00572 0,84840,00245 0,38070,00127 0,26980,00798 1,2450,01027 1,530,00777 1,1880,00416 0,6540,00101 0,167
Table B.4: Calibration diagram for G2 spectrometry.
Concentration wt % Absorbance
0,0443 0,5680,0088 0,1280,0013 0,0330,1028 1,242
B.0.4 Systems without lysozyme
Table B.5: Experimental data of ATPS system PEG6000 – dextran – water, for tie-line 1.
PEG6000 Dextran Dilution factor Composition (wt%)
Area Concentration Area Concentration PEG6000 Dextran Water
TOP 2,323 0,09166 0,9890 0,03310 0,0107 8,540 3,087 88,372,317 0,09144 0,9886 0,03308
BOTTOM 0,2398 0,01947 5,654 0,1936 0,0114 1,831 16,87 81,300,3244 0,02240 5,611 0,1921
B-2
Table B.6: Experimental data of ATPS system PEG6000 – dextran – water, for tie-line 2.
PEG6000 Dextran Dilution factor Composition (wt%)
Area Concentration Area Concentration PEG6000 Dextran Water
TOP 3,153 0,1204 0,4331 0,01397 0,0103 11,64 1,332 87,023,144 0,1201 0,4210 0,01355
BOTTOM 0,03372 0,01233 6,962 0,2386 0,0103 1,194 23,27 75,540,03198 0,01226 7,015 0,2404
Table B.7: Experimental data of ATPS system PEG6000 – dextran – water, for tie-line 3.
PEG6000 Dextran Dilution factor Composition (wt%)
Area Concentration Area Concentration PEG6000 Dextran Water
TOP 3,561 0,1346 0,3267 0,01030 0,0103 13,11 1,016 85,873,556 0,1344 0,3338 0,01055
BOTTOM 0 0,01116 8,545 0,2931 0,0116 0,9609 25,30 73,740 0,01116 8,585 0,2945
Table B.8: Experimental data of ATPS system PEG8000 – dextran – water, for tie-line 1.
PEG8000 Dextran Dilution factor Composition (wt%)
Area Concentration Area Concentration PEG8000 Dextran Water
TOP 1,320 0,05888 1,652 0,05590 0,0103 5,764 5,386 88,851,347 0,05980 1,625 0,05499
BOTTOM 0,3332 0,02462 3,546 0,1211 0,0103 2,523 11,79 85,680,4115 0,02734 3,566 0,1218
Table B.9: Experimental data of ATPS system PEG8000 – dextran – water, for tie-line 2.
PEG8000 Dextran Dilution factor Composition (wt%)
Area Concentration Area Concentration PEG8000 Dextran Water
TOP 2,369 0,09531 0,6324 0,02083 0,0102 9,290 2,024 88,692,355 0,09482 0,6258 0,02060
BOTTOM 0 0,01305 5,782 0,1980 0,0107 1,223 18,51 80,260 0,01305 5,756 0,1971
Table B.10: Experimental data of ATPS system PEG8000 – dextran – water, for tie-line 3.
PEG8000 Dextran Dilution factor Composition (wt%)
Area Concentration Area Concentration PEG8000 Dextran Water
TOP 2,991 0,1169 0,3466 0,01099 0,0103 11,39 1,088 87,522,995 0,1170 0,3569 0,01135
BOTTOM 0 0,01305 6,981 0,2393 0,0107 1,221 22,44 76,340 0,01305 7,020 0,2406
B-3
Table B.11: Experimental data of ATPS system G2 – dextran – water, for tie-line 1.
PEG Dextran Dilution factor Composition (wt%)
Area Concentration Area Concentration PEG Dextran Water
TOP 2,945 0,1774 0,2495 0,007650 0,0111 16,33 0,6442 83,033,046 0,1834 0,2187 0,006589
BOTTOM 0,1493 0,01283 7,512 0,2575 0,0106 1,287 24,42 74,290,1770 0,01446 7,590 0,2602
Table B.12: Experimental data of ATPS system G2 – dextran – water, for tie-line 2.
PEG Dextran Dilution factor Composition (wt%)
Area Concentration Area Concentration PEG Dextran Water
TOP 2,239 0,1359 0,4627 0,01498 0,0101 13,63 1,507 84,872,310 0,1401 0,4788 0,01554
BOTTOM 0,3581 0,02513 6,143 0,2104 0,0107 2,316 19,71 77,980,3464 0,02444 6,167 0,2113
Table B.13: Experimental data of ATPS system G3 – dextran – water, for tie-line 1.
PEG Dextran Dilution factor Composition (wt%)
Area Concentration Area Concentration PEG Dextran Water
TOP 5,508 0,2967 0,6307 0,02077 0,0203 14,97 1,023 84,015,766 0,3104 0,6297 0,02073
BOTTOM 0,4520 0,02717 13,43 0,4613 0,0202 1,574 21,08 77,340,6286 0,03658 11,43 0,3925
Table B.14: Experimental data of ATPS system G3 – dextran – water, for tie-line 2.
PEG Dextran Dilution factor Composition (wt%)
Area Concentration Area Concentration PEG Dextran Water
TOP 4,166 0,2251 0,1329 0,003637 0,0111 20,38 0,3352 79,294,225 0,2283 0,1382 0,003822
BOTTOM 0,1548 0,01133 8,383 0,2875 0,0099 1,064 28,91 70,020,1262 0,009804 8,362 0,2868
Table B.15: Experimental data of ATPS system G3 – dextran – water, for tie-line 3.
PEG Dextran Dilution factor Composition (wt%)
Area Concentration Area Concentration PEG Dextran Water
TOP 4,821 0,2600 0,9609 0,03213 0,0201 13,03 1,620 85,354,883 0,2633 0,9853 0,03297
BOTTOM 0,7063 0,04072 11,38 0,3908 0,0207 1,865 18,90 79,240,6286 0,03658 11,43 0,3925
B-4
B.0.5 Systems with Lysozyme
Table B.16: Experimental data of ATPS system PEG6000 – dextran – water with 0,3 wt% lysozyme, for tie-line1.
PEG6000 Dextran Dilution factor Composition (wt%)
Area Concentration Area Concentration PEG6000 Dextran Water Lysozyme
TOP 3,307 0,1258 0,3931 0,01259 0,0105 12,05 1,214 86,42 0,32003,371 0,1280 0,4041 0,01297
BOTTOM 0,05171 0,01295 7,880 0,2702 0,0116 1,095 23,31 75,28 0,31360,03702 0,01244 7,887 0,2704
Table B.17: Experimental data of ATPS system PEG6000 – dextran – water with 0,3 wt% lysozyme, for tie-line2.
PEG6000 Dextran Dilution factor Composition (wt%)
Area Concentration Area Concentration PEG6000 Dextran Water Lysozyme
TOP 3,693 0,1392 0,3164 0,009952 0,0108 1,172 24,89 73,63 0,30913,630 0,1369 0,3347 0,01058
BOTTOM 0,04249 0,01263 7,816 0,2680 0,0108 12,79 0,9511 85,95 0,30980,04292 0,01264 7,843 0,2689
Table B.18: Experimental data of ATPS system PEG6000 – dextran – water with 1 wt% lysozyme, for tie-line 1.
PEG6000 Dextran Dilution factor Composition (wt%)
Area Concentration Area Concentration PEG6000 Dextran Water Lysozyme
TOP 3,420 0,1297 0,3618 0,01151 0,0107 12,01 1,139 85,88 0,97143,371 0,1280 0,4031 0,01293
BOTTOM 0,04939 0,01287 7,979 0,2736 0,0116 1,120 23,64 74,17 1,0740,05534 0,01307 7,980 0,2737
Table B.19: Experimental data of ATPS system PEG6000 – dextran – water with 1 wt% lysozyme, for tie-line 2.
PEG6000 Dextran Dilution factor Composition (wt%)
Area Concentration Area Concentration PEG6000 Dextran Water Lysozyme
TOP 3,738 0,1407 0,3298 0,01041 0,0108 13,05 0,9323 84,92 1,0953,778 0,1421 0,3118 0,009794
BOTTOM 0,01498 0,01168 8,308 0,2849 0,0113 1,069 25,25 72,58 1,1020,03828 0,01248 8,330 0,2857
Table B.20: Experimental data of ATPS system PEG8000 – dextran – water with 0,3 wt% lysozyme, for tie-line1.
PEG 8000 Dextran Dilution factor Composition (wt%)
Area Concentration Area Concentration PEG Dextran Water Lysozyme
TOP 2,164 0,08818 0,7145 0,02345 0,0132 7,159 1,689 90,84 0,31232,542 0,1013 0,6511 0,02125
BOTTOM 0 0,01305 5,732 0,1969 0,0116 1,122 16,88 81,74 0,25610 0,01305 5,697 0,1957
B-5
Table B.21: Experimental data of ATPS system PEG8000 – dextran – water with 0,3 wt% lysozyme, for tie-line2.
PEG 8000 Dextran Dilution factor Composition (wt%)
Area Concentration Area Concentration PEG Dextran Water Lysozyme
TOP 3,420 0,1318 0,4152 0,01310 0,0102 12,85 1,262 85,58 0,30493,380 0,1304 0,4023 0,01265
BOTTOM 0 0,01305 7,572 0,2606 0,0106 1,237 24,72 73,79 0,25820 0,01305 7,586 0,2610
Table B.22: Experimental data of ATPS system PEG8000 – dextran – water with 1 wt% lysozyme, for tie-line 1.
PEG 8000 Dextran Dilution factor Composition (wt%)
Area Concentration Area Concentration PEG Dextran Water Lysozyme
TOP 2,558 0,1019 0,5773 0,01870 0,0112 9,080 1,616 88,23 1,0762,542 0,1013 0,5415 0,01746
BOTTOM 0 0,01305 5,760 0,1979 0,0105 1,239 18,63 79,11 1,0210 0,01305 5,659 0,1944
Table B.23: Experimental data of ATPS system PEG8000 – dextran – water with 1 wt% lysozyme, for tie-line 2.
PEG 8000 Dextran Dilution factor Composition (wt%)
Area Concentration Area Concentration PEG Dextran Water Lysozyme
TOP 3,098 0,1206 0,3329 0,01025 0,0112 11,01 0,9509 87,06 0,97813,134 0,1219 0,3459 0,01070
BOTTOM 0 0,01305 6,885 0,2368 0,0108 1,207 21,84 75,89 1,0640 0,01305 6,849 0,2355
B.0.6 Partitioning data
Table B.24: Experimental lysozyme concentration used to calculate the partition coefficients for systems withPEG 6000 - first measurement.
System S61Atop S61Abot S62Atop S62Abot S61Btop S61Bbot S62Btop S62Bbot
Abs 1,003 0,734 0,333 0,344 1,053 0,729 1,001 0,939Calc. Conc. 6,62E-03 4,75E-03 1,97E-03 2,05E-03 6,96E-03 4,72E-03 6,60E-03 6,17E-03
Dilution 33,2908 65,0054 134,1150 135,8805 149,7043 198,7671 158,7876 152,5725Phase conc. 0,258 0,271 0,264 0,278 1,049 0,947 1,049 0,942
Table B.25: Experimental lysozyme concentration used to calculate the partition coefficients for systems withPEG 6000 - second measurement.
System S81Atop S81Abot S82Atop S82Abot S681Btop S81Bbot S82Btop S82Bbot
Abs 0,294 0,343 0,260 0,312 1,053 0,729 1,133 0,758Calc. Concentration 1,70E-03 2,04E-03 1,47E-03 1,83E-03 6,96E-03 4,72E-03 7,52E-03 4,92E-03
Dilution 151,7738 132,5944 129,0917 111,4261 149,7043 198,7671 139,1991 183,6007Phase concentration 0,258 0,271 0,189 0,204 1,043 0,938 1,047 0,903
B-6
Table B.26: Experimental lysozyme concentration used to calculate the partition coefficients for systems withPEG 8000 - first measurement.
System S81Atop S81Abot S82Atop S82Abot S681Btop S81Bbot S82Btop S82Bbot
Abs 0,325 0,353 0,281 0,332 0,873 0,95 0,767 0,679Calc. Concentration 1,92E-03 2,11E-03 1,61E-03 1,97E-03 5,72E-03 6,25E-03 4,98E-03 4,37E-03
Dilution 145,6769 132,7104 149,995 146,454 182,4181 158,8730 206,466 215,756Phase concentration 0,279 0,280 0,242 0,288 1,043 0,993 1,02854 0,943155
Table B.27: Experimental lysozyme concentration used to calculate the partition coefficients for systems withPEG 8000 - second measurement.
System S81Atop S81Abot S82Atop S82Abot S681Btop S81Bbot S82Btop S82Bbot
Abs 0,301 0,333 0,318 0,346 0,888 0,722 1,01 0,919Calc. Concentration 1,75E-03 1,97E-03 1,87E-03 2,06E-03 5,82E-03 4,67E-03 6,67E-03 6,04E-03
Dilution 151,9428 146,2367 139,3086 145,4591 179,0995 205,8147 154,7906 154,3896Phase concentration 0,266 0,288 0,260 0,300 1,042 0,961 1,032 0,932
Table B.28: Experimental partition coefficients and deviation for the ATPS formed by linear polymers: PEG6000 – dextran – water and PEG 8000 – dextran – water. The partition coefficients were calculated with the
data from Tables B.24, B.25, B.26 and B.27.
S61A S62A S61B S62B S81A S82A S81B S82B
First measurement 1,020 1,051 0,899 0,899 1,003 1,191 0,952 0,917Second measurement 1,040 1,075 0,902 0,863 1,084 1,153 0,922 0,903Average 1,030 1,063 0,901 0,881 1,044 1,172 0,937 0,910Deviation 0,010 0,012 0,002 0,018 0,057 0,027 0,021 0,010
B-7
B-8